Not if "believe" from the followers is some kind of cosmic energy that powers the gods or defines their hierarchy, a pandimensional twitter subscription counter of sorts. In that case the last thing you want is to prove your existence as a god, because as soon as your existence is proven all believe will pop out of existence and will be replaced with knowledge of the existence.
Believe is only possible when the thing to believe in is vague and unknowable, you can't by definition ever believe in the existence of anything that you are sure of.
Oh no, I never said that they are only separable by time but that one is the predecessor or root of the other and that religions can spawn new cults of their own.
It can't only be time that transforms the one into the other, because otherwise we would have 100th of religions right now from all the sects and cults that have existed in the past.