I was dashing off a response to the original post when I noticed uvj's post, which I tend to agree with.
WRT your question above: India goes by multiple names: India (some theorize that it is derived from the river Indus, now a part of Pakistan, and home of the great Indus valley civilisation), Hindustan (stan literally means 'place' or 'location'), Bharat. In common usage Indian does indeed mean someone of the region. Amongst Indians, we tend to use the colloquial term 'desi' (which Sujeet mentioned, a while ago, has or had a pejorative context; something I was unaware of) to mean someone from the homeland/region (could include Indians, Pakistanis, others). I have even had friends differentiate between assimilated Westerners and the touristy or expat-club crowd by calling them former 'desi' and the latter 'phirangi' (foreigner), but I think this may not be general.
You are quite right about the languages (22 or so official languages, a lot more in reality, and 10s of dialects of each), enthnicity, etc. Before the British, there was not much of a unified India (though that claim is a bit misused, since vast territories were under the rule of Ashoka, the Moghuls, others).
Indian nationalism (and identity) is AFAIK a fairly recent concept, helped no doubt by sports teams and perceived enemies (Pakistan).
On an utterly unrelated note, it was interesting to hear Chomsky mention India's war against Pakistan (in aid of Bangladesh) as one of only two humanitarian military actions that he could think of (I think the other was Vietnam's entry into Cambodia?).
Someone with a less personal and more general knowledge of the history of India can probably correct any errors above and expand on this.
--ravi
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