“Anyone who says something like that obviously has little clue what a detonator actually is.”
Funny, when I worked at Precision Blasting (2006), the thing that was called a “Detonator” was the hand held battery operated push button detonator.
The blasting cap was always referred to as a blasting cap.
You may have called it a detonator, but you were using the wrong terminology. The detonator is the proximate device that causes detonation, and what you are referring to is simply a trigger.
Now, if we were talking low order explosives, it is possible for something to be both a trigger and a detonator, but not with high order explosives. Just doesn’t work that way.