Hindu as a geographical concept
The question what is yoga and is it a religion reflects the related question as to what is Hinduism and is it a religion?
Some say that Hindu is only a geographical concept and connotes anything Indian, which would quickly end any discussion of Hinduism as a religion. This idea, however, clearly does not work.
Hinduism is a legally-recognised religion throughout the world and has benefits accordingly, like any other religion, including the right of marriage, non-profit status, and the right to establish religious institutions. Those regarding themselves as Hindus in terms of religious identity constitute the most affluent and educated religious group in America along with the Jews, as recent studies have indicated.
Obviously we cannot limit Hindu as an identity to those born in India, as then Hindus outside of India, including Westerners who have taken on a Hindu religious identity, could not be Hindus. Note the Western-published magazine Hinduism Today, which comes out of Hawaii and is written mainly by non-Indians.
While Hindu may have originally been primarily a geographical term, it definitely has taken on a broader religious and cultural meaning. Mahatma Gandhi himself used the term in a religious sense and regarded himself as a Hindu.
Hindu as Sanatana Dharma
If one looks at the ancient literature, the correct term for Hinduism would be "Sanatana Dharma" or the Eternal Dharma, with Shaivite, Vaishnava, Shakta and other groups as its branches. This would identify Hinduism with the Vedic tradition, but in a flexible manner. Most people in the world today tend to do this, regarding, for example, the Bhagavad Gita as the main scripture of Hinduism, though the Gita as a text is older than the term Hindu for Sanatana Dharma.
A case could be made for replacing the term Hinduism with Sanatana Dharma to reduce misconceptions. But, however preferable, that would take much time and effort to accomplish. Hinduism is a legally and academically-recognised religion in the world and the term Sanatana Dharma is not well known.
Yet Hinduism is not a religion like the dominant Western religions, which complicates the discussion. It has no single or final prophet, One God, scripture, institution or code of belief. This has caused some people to say that Hinduism is not a religion at all.
However, Hinduism has probably the world's largest and oldest spiritual and religious literature through the many vedas, puranas, shastras, tantras and modern gurus. It has numerous temples, deities and monastic orders, probably more so than any other religion. Hindu religious festivals like the kumbh mela are by far the largest in the world.
We can perhaps better define Hinduism as a pluralistic religious, cultural and spiritual tradition, rather than try to limit it to the terms that Western religions originally made for themselves. Buddhism is not much different in this regard and conforms even less to the usual Western idea of religion, as it has no God or creator, which many Hindu sects do.
Hinduism as more than a religion
Some say that Hinduism is a culture or way of life, which has some validity as well. Hinduism includes many aspects of culture like art, music, dance and literature. In addition the term can refer to a way of life like the rules of daily living that many Hindus follow.
Some regard Hinduism like yoga as a science of consciousness. Certainly yoga practices of various types are common in all branches of Hinduism. Some prefer the term Hindu Dharma over Hinduism, to bring out more the connection with dharma.
So rather than stereotype Hinduism or Sanatana Dharma in simplistic verbal definitions, perhaps we should just accept the term Hinduism or Hindu Dharma for now - but redefine it in a broader sense as a pluralistic tradition that encompasses religion, spirituality, philosophy and culture. Hindus can choose to practice any part of these, with different Hindu sects having their own emphasis. Even atheists are not barred from being Hindus if they accept the principles of Dharma.
There can be religious or non-religious Hindus, but Hinduism does have an important place among the great religions of the world, whatever name you prefer to give it.