[constantcrying]

>I wonder if there's a terminology schism here between computer graphics and numerical analysis folks.

The first group just pretends every function has a derivative (even when it clearly does not), the other doesn't.

The linked Wikipedia article gets it exactly right, I do not know why you would link to something which straight up says your definition is incorrect.

There is no point in talking about Lipschitz continuity when assuming that there is a derivative, you assume that it is Lipschitz because it is a weaker assumption. The key reason Lipschitz continuity is interesting because it allows you to talk about functions without a derivative, almost like they have one. It is the actual thing which makes any of this work.