On Hindu, Indian or Bhāratīya identity.

It is well known that the term Hindu originates from the Iranian version of Sanskrit term Sindhu, referring to water bodies such as rivers or specifically to the modern Indus river which is called Sindhu in Sanskrit. The earliest usage of the term Hindu is found in Iranian Avestan text of Vendidad 1.18, which is one of the sacred texts of Zoroastrians. This text mentions Hapta Hindu (read Heñdu) or Vedic Sapta Sindhu region, the land of seven rivers located in northern India. Behistun Inscription of Achaemenid Persian ruler Darius I who reigned in 6-5th centuries BCE also mentions the term Hindu (read Hiduš) referring to the regions in northern India. In later times, the term began to refer to the whole of Indian subcontinent and its inhabitants.

After the Achaemenids, the Greeks called the subcontinent region as Indós, India etc which is also derived from Sindhu in the same way how Hindu is derived from Sindhu.

So both the terms Hindu and India are synonymous, and have same geographical meaning referring to the civilization of Indian subcontinent and it's inhabitants.

The the term Hindu religion or Hinduism is used to collectively refer to all of the diverse non-Abrahamic Āstika traditions practiced in India, such as Vaiṣ‌ṇavism, Śaktism, Śaivism etc which accepts the authority of the Veda-s. There are also other Nāstika religious traditions like Buddhism, Jainism etc which do not accept the authority of the Veda-s.
Even when there existed intense rivalry between Āstika and Nāstika sects, they also shared many things in common. Concepts like Dhyāna, Mokṣa or Nirvāṇa, Dharma, Karma etc are common to both Āstika and Nāstika traditions, even though the interpretations may differ.
Also, many historical sites like Ajanta, Ellora, Elephanta, Badami, Khajuraho etc hosted both Āstika and Nāstika places of worship. They also used same symbols like Svastika, Śrīvatsa, Nandipada or Triratna, tree-in-railing etc during early historic period.
Āstika-s and Nāstika-s through many centuries of co-existence and inter mingling also would have influenced each other's diverse concepts and ideas.
So technically both Āstika and Nāstika traditions share same Dharmic heritage. It is upto the Nāstika-s to decide whether to call themselves as part of larger Dharmic Hindu umbrella along with the Āstika-s.

As for the tribal and folk religious traditions, majority of them are also influenced by either Āstika or Nāstika traditions.

So in sum, Hinduism being based on a geographical or civilizational term and consisting of various diverse traditions, would be more than just a religion and can collectively refer to the diverse civilizational traditions of Indian subcontinent.

Now some say the terms Hindu and India are not indigenous terms, and hence Hinduism or India didn’t exist before foreigners gave these names to the subcontinent and it’s people. Like pointed out before, both of these terms are foreign translations of Sanskrit term Sindhu.
So why shouldn't we call it by a Sanskrit term itself?

The native Sanskrit term of our civilization is Bhārata or Bhāratavarṣa.
The unity of the civilization of Bhāratavarṣa for the last 2000 years is visible from the early Caṅkaṁ era Tamil texts like Patiṟṟuppattu 11 and 43 which treats the regions from northern Himalaya to southern Tamil land of Kumari as one unified entity. Caṅkaṁ era is usually broadly dated from 500-300 BCE to around 300 CE.
The unity of Bhāratavarṣa is also evident from the Hāthigumpha inscription of Kaliṅga king Kharavēla dated around 200-100 BCE, in which Kharavēla talks about his expedition against the other kingdoms of Bhāratavarṣa. So Bhāratavarṣa was a civilization with many kingdoms within.

The name Bhāratavarṣa is derived from the name of the most illustrious Vedic kings, Bharata Dauhṣanti. He was son of king Duhṣanta (alternatively as Duṣyanta, Duṣmanta etc) and his wife Śakuntaḷa, from the Puru dynasty of larger Aila Candravanśa or lunar lineage of Vedic Kṣatriya-s.

As per Sambhava Parva of the great epic Mahābhārata, the Bhārata-s, most prominent tribe mentioned in the Ṛg Veda, the earliest Indian text, were also the descendants of this heroic king Bharata Dauhṣanti.

Glorious kings of Ṛg Veda like Sudāsa and Divōdāsa was from this tribe of Bharata and they were victorious in important battles mentioned in Ṛg Veda like Daśarājñá. Bharata Dauhṣanti is also mentioned in other Vedic texts such as Aitareya Brāhmaṇa 8.23 and Śatapatha Brāhmaṇa 13.5.4.14 , where it is stated that Bharata had attained the greatness which neither the men before nor those after him had attained.

So I feel it is better if we call our civilization (more than mere nation) as Bhārata or Bhāratavarṣa, and ourselves as Bhāratīya-s , a name which reminds of the civilizational patriarch Bharata Dauhṣanti.

It is also to be noted that the term Bhārata is also used as an epithet of Agni, for example in Ṛg Veda 4.25.4. Agni is the sacred fire representing the light of Vedic culture and without it's use no Vedic ritual was possible. Thus the term Bhārata is also connected directly with Vedic tradition, which is the foundation of our later traditions.

Since our great heroes like Chhatrapati Śivāji also utilized the term Hindu, I see nothing wrong in continually using it, although for an autochthonous identity, we can call ourselves as Bhāratīya-s.

The Abrahamic traditions and its followers of course are not part of this Bhāratīya or Indian or Hindu identity, since Bhāratīya traditions predate Abrahamisms and would collectively refer to the non-Abrahamic Dharmic traditions of Bhāratavarṣa derived from or influenced by the Vedic tradition and its later developments.

Vedic notion of divinity

Currently in Āstika Dharmic traditions which accepts the authority of Veda-s, we have different sects worshiping different Devata-s or deities as supreme. For example the Śakta-s worship Śakti or Devi, Vaiṣṇavas worship Viṣṇu, Śaivas worship Śiva etc as their supreme deities. During the Vedic era, there was more emphasis on the worship of deities like Agni, Indra, Varuna, Mitra, Vayu, Surya, Aditi, Savitṛ, Tvaṣṭṛ , Soma, Aśvini Kumara-s, Yama etc. Many of these divinities got their status changed in the later post-Vedic Āstika traditions when Viṣṇu , Śiva , Śakti etc got prominence.

The Vedic hymns are dedicated to a particular deity or sometimes group of deities in which the respective deities are praised as the supreme. For example in Ṛg Veda 1.98.1 Agni is praised as the supreme deity, in Ṛg Veda 6.36.4 Indra is the supreme, in Ṛg Veda 7.99.2 Viṣṇu is the greatest, in Ṛg Veda 2.33.9 Rudra is the mightiest, in Ṛg Veda 9.42.2 Soma is deity above all other deities, in Ṛg Veda 10.170.3 Surya is the supreme etc. So each Vedic deity has its own importance and is worshiped as the supreme deity in their respective hymns.

But besides this, there was also an abstract concept of the supreme absolute during the early Vedic era. Although it was not as popular as the other divinities, we can find reference of the absolute one or Eka in various scattered Ṛg Vedic hymns like Ṛg Veda 10.121, 10.129, 3.54.8, 10.81, 10.82, 3.55, 1.164.6, 1.164.46 etc. This Eka can be considered as an abstract state which represents totality of all the deities or all the deities reflect multiplicity of this abstract entity. Ṛg Veda 1.164.46 specifically mentions Ekam Sat or one being which the sages describe in multiple ways. This abstract concept of Eka or one entity, in my opinion, is precursor to the later concept of absolute Brahman. Any one who is familiar with the Āstika philosophies would also be familiar with the concept of Brahman, the changeless, eternal, absolute cause or source of the universe or existence and Ātman, our true self which is identical with supreme Brahman. In the Vedas, the term Brahman signifies prayer, devotion or the power of Mantra-s etc. In Ṛg Veda 4.16.9 the term Abrahma meaning unholy, prayerless etc also occurs. Brahman is also closely associated with Vak or sacred speech representing Vedic hymns, for example in Ṛg Veda 1.164.35. So in my opinion the absolute Eka Brahman can be termed as the mystical holy source or cause of the universe or existence.

I shall now explain this concept of the Eka or Brahman as found in the Vedic texts. Although there are many interpretations about this concept in various texts, I will mainly detail based on what is explained in the early Vedic texts. There are many names and aspects of Eka or Brahman in the Vedic corpus. It is called on as Hiraṇyagarbha, Prajāpati, Viśvakarma, Puruṣa etc.

Prajāpati - Prajāpati or lord of hosts (i.e created beings) is the main creator in the Vedic texts. Prajāpati first appears in late parts of Ṛg Veda and becomes a prominent figure from later Yajur Veda onwards. The name is also identified with many other deities but it is frequently applied to the abstract creator who is mentioned in Hiraṇyagarbha hymn from Ṛg Veda 10.121 and which represents all other deities. Prajāpati is obviously the creative aspect of manifested Eka or Brahman and in Śatapatha Brāhmaṇa 7.3.1.42, a Vedic ritualistic text, he is explicitly identified with the whole Brahman representing the totality or the omnipresence of manifested existence. In Śatapatha Brāhmaṇa 2.1.2.13 it is stated that all other deities sprang from Prajāpati and Prajāpati also represents all the deities according to Taittirīya Samhita 2.1.4.3 Yajur Veda, reaffirming the fact that Prajāpati is an abstract figure which unifies all the deities. Prajāpati also has a very important role in the Vedic Yajnas or sacrifices. Prajāpati , although viewed as a father figure, also has motherly aspects. Prajāpati has a womb in Śatapatha Brāhmaṇa 8.2.2.5, becomes pregnant with all the creatures in Śatapatha Brāhmaṇa 8.4.2.1 and also has milk yielding breasts in Pañcavimśa Brāhmaṇa 13.11.18. Further in in Śatapatha Brāhmaṇa 5.1.5.26 it is stated that Prajāpati is both father and mother, representing father sky and mother earth. Prajāpati is clearly an enigmatic figure. Perhaps, the Vedic authors wanted to appropriate motherly female elements into the fatherly male Prajāpati , to make him as close as possible to the absolute brahman which has no gender.

In post-Vedic texts, the status of Prajāpati got degraded and there were many different Prajāpati-s. Prajāpati 's creative role also became limited to the creator deity Brahmā of Trimurtis. Also Avatara-s of Viṣṇu like Kūrma (tortoise) and Varaha (boar) in post Vedic texts are also associated with Prajāpati in Vedic literature. For example Śatapatha Brāhmaṇa 14.1.2.11 mentions Prajāpati taking form of boar to lift up the earth just like Varaha Avatara and Śatapatha Brāhmaṇa 7.5.1.5 also mentions about Prajāpati taking form of tortoise, reflecting Kūrma Avatara.

Puruṣa - Puruṣa , also called as Nārāyaṇa, is the cosmic male or personified universe. Puruṣa is frequently identified with Prajāpati like in Śatapatha Brāhmaṇa 7.4.1.15 and has similar attributes like Prajāpati . Puruṣa also plays an important role in many Vedic sacrifices. According to Puruṣa hymn from Ṛg Veda 10.90 which re-occurs in many other places with slight modifications, the whole world was manifested from the body parts of Puruṣa by the sacrifice of Puruṣa performed by the Gods. Puruṣa represents multiplicity of the universe and has multiple body parts. Puruṣa is everything, the entire manifested existence itself represented both time and space. Puruṣa fills the whole space in Ṛg Veda 10.90.1 and he is the past and the future representing time in Ṛg Veda 10.90.2. Atharva Veda 10.2.30 also identifies Puruṣa with Brahman.

In post-Vedic texts, Puruṣa Nārāyaṇa is identified with Viṣṇu and the multiplicity of Puruṣa became an important aspect of Viṣṇu . Puruṣa also became associated with Śiva since Linga or phallic symbol representing Puruṣa also became symbol of Śiva in later traditions. Puruṣa also plays an important role in Sāṃkhya philosophy, where he represents the supreme soul as opposed to his female counterpart named Prakṛiti or the manifested universe.

Viśvakarma - Viśvakarma the all doer or all worker is another abstract state like Puruṣa-Prajāpati and is identified with Prajāpati several times like in Śatapatha Brāhmaṇa 8.2.3.13. Just like the name Prajāpati, the name Viśvakarma is also applied to various deities, but it is mostly used for the abstract creator who is mentioned in Viśvakarma hymns of Ṛg Veda 10.81 and 10.82. These hymns also occurs in various other Vedic texts. Like Puruṣa, Viśvakarma also represents multiplicity and has many organs as mentioned in Ṛg Veda 10.81.3.In post-Vedic texts, Viśvakarma was viewed as heavenly architect.

Hiraṇyagarbha - The name literally means golden fetus, womb, embryo, germ etc .It is also possibly identical to Hiraṇya aṇḍa or golden cosmic egg mentioned in various Vedic texts such as in Śatapatha Brāhmaṇa 11.1.6.1. This concept first appears in the Hiraṇyagarbha hymn of Ṛg Veda 10.121 where it is praised as the supreme absolute. This hymn is repeated many times throughout the Vedic texts just like Puruṣa and Viśvakarma hymn. Hiraṇyagarbha is also called on as Ka or who, an unknown mysterious entity, and is identified with Prajāpati in many instances like Ṛg Veda 10.121.10, Śatapatha Brāhmaṇa 6.2.2.5 etc. Vājasanēyi Samhita of Śukla or white Yajur Veda 32.4 speaks about unity of everything in context of Hiraṇyagarbha . As the name indicates, Hiraṇyagarbha refers to the root cause or the infant state of absolute Eka or Brahman from which Prajāpati-Puruṣa-Viśvakarma representing everything, the whole manifested existence, arose. Thus, it is the closest state to the changeless and eternal unmanifested absolute Brahman.

So to sum up, Prajāpati-Puruṣa-Viśvakarma refers to same creative or manifested aspect of the absolute Eka or Brahman which represents the existence of the universe. Hiraṇyagarbha on the other hand refers to the root cause of existence which is beyond existence and thus represents the unmanifested state of absolute Eka or Brahman which is not yet fully existent.

There are also other abstract states which are not as popular as the above mentioned concepts in Vedic religious thoughts. For example, Skambha and Rohita which occurs in Atharva Veda 10.7 and 13.1. Skambha is a cosmic pillar which supports the universe and Rohita is symbolized red shining Sun which represents the absolute.