The problems are rather the same: is caliing David an English name rather than, say, French or Hebrew better than calling it a Christian name rather than, say, Jewish? You can think of India like you think of Europe and the middle east combined.
Note that the pronounciation, but often not the spelling, changes from one language to the next, a slightly confusing concept, since for about a thousand years the scripts are mutually unintelligible, but the alphabets *do* correspond. So, in pronounciation questions, the language is essential.
In most other contexts, I think we just have to be vague: just like an unqualified use of the term Spanish name somewhat implies that it is not restricted to the Basque population, and that it is probably not used in France; I suggest calling most of these names Indian. When a more specific usage is known, we can put it in: like North Indian name. Sometimes, it may be known to the language like a Hindi or Bengali name, and, yes, sometime to a religion like a Parsee name. Conversely, even though there are a lot of Muhammads in India, we should not call it an Indian name.
Sorry to take up board space with such trivialities: but a Hindu name does jar!