Atheism: the gods do not exist, regardless of human beliefs.
Dei ex hominibus: the gods do not exist if humans do not believe in their existence, they exist if humans believe in their existence. Thus, in some ways, humans create gods, and also destroy gods by no longer believing in them.
Potestas deorum ex hominibus: the gods exist regardless of human beliefs, but they only have power (over humans and/or in general) if humans believe in their existence. A possible interpretation is that the gods get their power from the psychic energies of humans, acting in some ways as psychic vampires. It could be that some gods (e.g. those that are in control of natural phenomena, like thunder, rain, earthquakes etc) have power regardless of human beliefs, but some other gods (e.g. those that prescribe a specific set of rules not found in nature, like Yahweh, Allah etc) are psychic vampires that only have power when humans feed them with their (the humans') psychic energies, and when humans no longer believe in those vampiric gods, then those vampiric gods, while still existing, lose all their powers and become weak and almost moribund (though never dying), waiting in the hope that humans will give them their psychic energies again. Or it could also be that both categories are psychic vampires, and that in an atheist society, natural forces are really not controlled by any deity.
Dei Terrae ex hominibus: similar to 'potestas deorum ex hominibus' in that human belief in the gods change one of their characteristics, but here, instead of the amount of power, what changes is their 'location' so to speak - the gods exist regardless of human beliefs, but when humans do not believe in a particular deity, that deity remains in a metaphysical realm, unable to affect our world, whereas if humans believe in a particular deity, then the psychic energies coming from humans act as a way to 'summon' that deity into our world. When humans stop believing in that deity, that deity leaves Earth and returns to the metaphysical realm, waiting to be summoned again. That deity does not necessarily need to be believed in by the entire society in order to be summoned, it can also be believed in by a small sub-culture, or even just an individual. Of course, in that case, the deity is 'weaker' and has less control over our world or society, because of the lower amount of psychic energies that deity can use to empower itself.
Voluntas deorum ex hominibus: here, human belief in the gods determines whether or not those gods will affect human society. Unlike in "Potestas deorum ex hominibus", in this interpretation the gods can affect human society regardless of human belief, but they consider it pointless if the humans do not believe in that god's existence. In this interpretation, the gods interpret human belief as desire rather than "belief" proper. When a human society starts believing in one or more gods, those gods interpret it as a desire from humans, perhaps a call for help, and so those gods start intervening. When a society stops believing in certain gods, then those gods will interpret the lack of belief as "we are no longer needed, welcomed or desired here", and so they'll abandon that society.
Cognitio deorum ex hominibus: the gods exist and have power over humans regardless of human beliefs, humans can only either be ignorant of the gods and their power, or be aware of them. This, along with 'atheism', is one of the two options (so far) where not only the existence of the gods but their characteristics as well do not change based on human beliefs.