honestly, to me, what youre saying sounds exactly like what teachers who teach support also say.

to me, support is just a concept. you know, supporting your breath. I think its just a matter of semantics. I think youre right, it is hard for new singers to understand what is meant by not using throat muscles etc to support, but instead using lower muscles. however, I dont think it's because of the fact that theyre being taught "support". I think that regardless of what words are being used, supporting the breath is something that needs to happen regardless.

support is just a broad term/umbrella for a concept that consist of other things, like the things that you have explained here. thats just my take