Here is my two cent if the "other" gods had gotten their power from anywhere but through worshipping.
By the heavens and earths, they shall be NATURAL gods. As in the one who birthed them is the nature itself, which is weird considering that THESE GODS should be the first ones to appear than the FAITH gods. If they appear earlier, then they should be called OLDEN gods. So the power associated should be Aether?
By themselves or through means that resulted in them accumulating the energy, be it from getting struck by lightning or having to cannibalise their way through. These are called EARTHEN gods or MORTAL gods, as they stem from the filthy roots that is mortality itself. Gods like these include living beings like animals, humans (split because "humans are special"), plants to even things like inanimate objects* that somehow gained sentience and start absorbing the energy by themselves. Ehm, I just throw in Mana here.
*Quick sidenote, the difference between inanimate objects turning into either FAITH gods or MORTAL gods is whether the energy has been fed to them through other means or if the object itself had seek the energy by itself.