You can see from looking at names like 'Indonesia' and 'The East Indies' that the word 'India/n' was not exclusively applied to the Indian subcontinent by Europeans of the period, even within Asia itself. Rather 'India' referred to a large, ill-defined and ill-understood area.

Although it was realised quite soon after Columbus that he had not discovered 'the real india', the term was vague enough that they could be described as 'the west indies' (as parts of the Caribbean still are today), in contrast to 'the east indies'.

Here is an explanation of what 'India' meant in the European imagination, although I'm not sure where he's getting his sources from.