No, this is not the fundamental meaning of an index, it is simply the interpretation that you are using to make it easier to deal with for you personally. In my opinion, the fundamental meaning of an index is a number referring to element number `n`, so that `a[1]` is the first element, etc. But I don't begrudge you your working definition.

For some reason, people who are used to 1-based indexing generally recognize that both interpretations are perfectly fine, and that each works well in different contexts. For some other reason, people used to 0-based indexing feel so intellectually superior that they are unable to see the other side of the argument.