Yeah that is one way of doing it, but I'm not sure how practically useful it actually is; many functional languages do allow for mutation and/or side effects too. Often in a controlled way, of course, but then again, so does Rust. That said, I wouldn't argue that Rust is a functional language exactly, mostly that I find essentialist FP definitions to be lacking, like most essentialist definitions.

That said, you're right that "higher order" is a better description of these features, but my counter would be that most people perceive higher order programming as an aspect of functional programming, bringing it full circle again :)