In section IB3.5, beginning on page 186 of The Fundamentals of Mathematics, Volume 1, the terms order and power of a finite group are used. In the first part of the book the power of a finite group is said to be its cardinality which, for a finite set is the whole number of elements in the set. That is what I like to call the size of the set. I'm not sure what to make of
the order of a subgroup of a finite group $\mathfrak{G}$ is always a factor of the order of $\mathfrak{G}$.
I believe it simply means the number of elements in a subgroup of $\mathfrak{G}$ will always divide evenly into the number of elements in $\mathfrak{G}$.
At this point we apparently have index, size, order, power and cardinality used to indicate the number of elements in a set under specific condition.
Am I correct in understanding that power and order mean the same thing in this context, and that thing is the number of elements in the set?
I've not heard "index" used to describe the cardinality of a group. The index of a subgroup of a given group is the number of cosets of the subgroup (or the size of the quotient group, in case the subgroup is normal), but that's not the same thing.
Meanwhile, while I've heard "power" used to mean the cardinality of a set, I've not seen it used that way around groups. It's also an older term, and modern texts don't use it as far as I can tell.
One does need to be a bit careful about "order" - while the order of a group (finite or otherwise) is indeed its cardinality, we also talk about the orders of an element of a given group (= the cardinality of the subgroup generated by that element). So e.g. we can have a group of order $\aleph_0$ with elements of order $17$.
So tl;dr: with the exception of "index" (as far as I know) they do mean the same thing, but "power" is archaic and mostly set-theoretic and I'd avoid its use in all contexts.