> The fallacy itself doesn't constitute a counter argument.

It isn't a reason to believe that the position argued for is false, but it is a reason for rejecting the offered argument as a basis for believing that the positioned argued for is true.

I've heard "counter-argument" used both to mean "argument for rejecting the offered argument" (which this satisfies) and "argument for the negation of the position the offered argument supports" (which this does not satisfy.)