[thatmathguy]

>Textbook authors let some categorical language slip through so the time, and I think eductors are doing students a disservice my neglecting it.

I agree. When teaching, I try to choose my terms carefully so someone would not go astray from the subject matter. It is remarkable how easy it is to get stuck on a side-quest with looking up abstract terms, while-so missing the main topic. imo the graceful way to handle it is to drop references (and disclaimers in case they are time-consuming) when introducing new things.

Somehow math learning has the property that 'struggle with x, finish x. learn y. if I knew y before x it would've been so much smoother!' sounds correct, but in practice it often does not go so well.