Right, it does depend on this perspective, which is why I say the distinction is often lost in practice. But from the perspective of hash table designers (at least historically), the hash can in fact change and is dependent on the size of the table. The distinction is important because the two operations are separable. I guess people didn't want to call one thing a "hash" and the other "hash mod k" because there are ways to do this other than mod. They could have called one "hash" and the other "post hash" but it is what it is.

All I'm trying to say is if you want to know more about hash tables, and you bother to venture into the literature or some textbooks, you're going to run into this distinction sooner rather than later, so I wanted to point that out here.