I've heard omnibar used to describe the more general UX pattern of providing at a focal point a single freetext input that surfaces items across collections. It's a neat pattern because it's both pretty and can be useful.

I'll take your word for it it came from Chrome's Omnibar, the timeline in my memory lines up.

TIL, thanks! I've added that word.

(In my defense, even the support.mozilla.org docs themselves seem to avoid it; the one I linked only refers to it as "address bar").

Avoid awesomebar? That is a nickname - the correct term is the address bar. It has never been an "omnibar", however.