Why do some openings have Indian in their name? Why are a few chess openings named King's Indian Defense, Queen's Indian Defense, Nimzo-Indian defense, and so on?

What is the significance of Indian in all these opening names? What's India's contribution to it?
In Indian Chess, the game that was played in the 18th, 19th and early 20th centuries in India (not to be confused with its ancestor Chaturanga), the rules allowed no castle, but :

The king can make a knight's move once in a game, known as Indian castling.

As a consequence, g3 followed by Kg2 for White or ...b6 followed by ...Kb7 (the black king stands on d8 at the beginning of the game in Indian chess) were very common moves in the opening, since it was the fastest way to bring the king towards the corner where it will usually find the most safety.
Of course, playing g3, b3 or ...b6, ...g6 could also be followed by a &quot;normal&quot; fianchetto with the camel (our bishop).
When Indian players started playing western chess (during colonization), they quite naturally used a lot of fianchettoes, since they are common in Indian Chess. As explained in user1583209's answer, that was the case for Moheschunder Bannerjee in his games against Cochrane around 1850, and The Chess Player's Chronicle used the term &quot;Indian defense&quot; to refer to 1.e4 d6 2.d4 g6 after him, about 30 years later though (1884). This is the first documented use of the appelation about that kind of defense with fianchetto - although &quot;Indian Opening&quot; appeared earlier to refer to 1.e4 e5 2.d3.
The term &quot;Indian Defense&quot; as we understand it today (After 1.d4 Nf6, thus excluding Pirc-Ufimsev and other schemes with e4 and not c4 for White) established itself after WWI, its use being popularized by Savielly Tartakower and Hans Kmoch.