In Hinduism we have monotheists as well as polytheists. We do not have same deities and same customs. Does one become Hindu by accepting supremacy of Vedas? We don't have a single book like other monotheists religions. Or is Hindu a person who believes in theory of karma and concept of moksha. Or does a Hindu believe in caste system. There are some Hindus who have concept of pure and impure. Then we have some Hindus who say that the world is god's manifestation and there is nothing impure in it. So what makes a Hindu a Hindu.?
It is very difficult to term a person as Hindu, under a single method.
1. Accepting the Vedas as the authority
The earliest spiritual texts in India are Vedas. So if anyone says that a person shall be a called as Hindu, if he accepts the authority of the Vedas, then that claim is questionable.
In order to accept the above claim, the foremost requirement is whether that person understands the Sanskrit used in the Vedas, as the Sanskrit in Ramayana, Mahabharata, etc is different from that used in the Vedas.
Mere studying Vedas with the help of translated versions, is not a reliable method. What is the guarantee that the translator of the Vedas understood the core meaning of the Vedas, properly?
It was rumoured that scholars like Max Muller translated the Vedas with malafide intentions. (https://www.encyclopediaofauthentichinduism.org/articles/35_max_muller.htm ). So they are not dependable.
2. Accepting the theory of karma and concept of moksha or in caste system
A person may be interested only in SPIRITUAL practices, sans belief in any theory like Karma or cycles of rebirths. That person may follow the age-old traditions like receiving and feeding guests, telling truth, etc. Shall we call him a Hindu or not.
A debatable question.
How about Saint Kabir? (https://en.wikipedia.org/wiki/Kabir)
He prayed to Sri Raama all the time, but his birth into a Muslim family is doubtful. Was he Hindu or not?
3. Belief in Advaita philosophy
If belief in Advaita is a benchmark, then what about persons following dualism like Vaishnavism, Shaivism, etc (human soul and God are different)?
Meaning of Hindu:
The historical meaning of the term Hindu has evolved with time. Starting with the Persian and Greek references to the land of the Indus in the 1st millennium BCE through the texts of the medieval era, the term Hindu implied a geographic, ethnic or cultural identifier for people living in the Indian subcontinent around or beyond the Sindhu (Indus) river.
By the 16th century, the term began to refer to residents of the subcontinent who were not Turkic or Muslims.
If we go a little back, earlier people used to call the traditions being followed by the people in this country as SANAATANA DHARMA, but not Hinduism.
SANAATANA DHARMA is a broad term, which also includes following ancient traditions, practicing SPIRITUAL concepts, etc.
In Ramayana, Sanaatana Dharma was used here and there.
कृते च प्रतिकर्तव्यमेष धर्मः सनातनः |
सोऽयं तत्प्रतिकारार्थी त्वत्तः संमानमर्हति || ५-१-११४
"When a service is done, a return service needs to be rendered. This is an ancient tradition. Such this ocean who wants to do a return service to the Raghu dynasty, is suitable for respect by you."
Sita mentions that one who does not get disburbed with sorrow or joy is blessed person. This is what even Bhagavad Gita says.
धन्याः खलु महात्मानो मुनयस्त्यक्तकिल्बिषाः || ५-२६-४९
जितात्मनो महाभागा येषां न स्तः प्रियाप्रिये |
"Great souled ones with abandoned sins, with a conquered mind, those with great fortune, sages are indeed fortunate to whom there is no pleasure and displeasure."
Ultimately, it boils down to these aspects, ie., following ancient traditions, practicing SPIRITUAL concepts, etc, which may be used as a benchmark, to decide whether a person is called HINDU or the follower of SANAATANA DHARMA.
The term 'Hindu' is recent. The name of our religion is Veda (Vaidika) dharma, or Sanatana Dharma, because it is based on Vedas.
Generally accepted philosophical definition is - Astika, which is belief in Vedas.
Generally accepted religious definition is - belief in Bhagavan, deities, scriptures (Gita, Ramayana, Mahabharata), rebirth (multiple past and future lives), karma.
Generally accepted cultural definition is - going to temples and celebrating festivals like Diwali, Ram Navami, Shivratri.
Generally accepted geographical definition is - Dharmic religions that originated in Bharata, possibly including Jain & Buddha.