Indeed. People in India call themselves Bharati and their country Bharat. "India" is the European name for the country, coined by Alex T. Great after the Indus Valley which he conquered.
Well, to be fair, that latter name was derived from terms used by the Persians (not quite European, themselves) who had made it to the Indus before the Two-Horned One showed up.
But even so, both India and Bharat are equally official names for the Republic.
And in India's case, that term, or a derivation of it, has been in use both in Asia and Europe for millennia, and if anywhere has more currency there than it does regarding the indigenous peoples of the Americas.
(Although, ironically, the civilisation of Harappa and Mohenjo-Daro was when the Indus-derived term was most apt - perhaps a more relevant term in later eras could have been a derivation of 'Ganges', given its emergence as the main artery of the country after the Indus Valley civilisation's collapse.)
Alternatively, one could call neither people 'Indian' and avoid that issue altogether... and probably end up opening a whole set of new ones.