Yeh a broiler here (Australia) is just as you describe and while I hadn’t given it much thought is too used in the mode you describe as well, it’s just that it’s called a “grill” and the act of using it is to “grill” which is an amusing point of confusion since it seems to have very different connotations in the states. I was aware that “Grilling” over there also connotated using the outdoor grated cooking surface that I’d call a “barbecue” but I guess where I was confused is that I thought the term also covered those otherwise flat surfaces that have the ridges like those George Foreman “grill” things but apparently that’s actually what a “griddle” is so that clears things up a bit. In either case I still can’t understand why a toasted cheese sandwich cooked in a frying pan gets called “grilled” and funnily enough it’s common to make a variation of that here that’s not quite as good but much easier and lazier to make where you put a single slice of bread covered with cheese (though not the American kind as that probably wouldn’t work very well with this method) under what I believe you’d call the “broiler”. This local method of melting cheese on bread really added to the confusion before I became aware that “Grilling” meant something different over there because I figured this must have been what was meant by “grilled cheese” before I figured that out lol.

I think this system of classifying sauteing vs frying, is quite useful, a bit more precise than what I’m used to, just doesn’t seem to get much use amongst my circles here. Still the lack of distinction necessarily made between degrees of “fried” is interesting since “fried” chicken seems to quite specifically mean deep fried even if for many dishes a person might well intend to use a lot of oil to cook some chicken but not necessarily plan to deep fry it.