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.