It's a philosophical proposition (theism) and it's philosophical antithesis (atheism). These terms never "applied to people". But when people express their allegiance with one or the other of these philosophical positions, we refer to them as "theists" or "atheists".
Those words refer to things people do or don't do. Not to things they assert as true or they reject as untrue. I'm not confused about that at all.
I find that when people fight to misuse words deliberately and repeatedly, even when they've been corrected, it's probably because they want to obscure the truth rather than reveal it.