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.
Vigilia deorum ex hominibus: the gods exist regardless of human beliefs, but when people don't believe in them (or have no concept of "gods" in their culture), the gods sleep. Once people start believing in one or more specific gods, they (the people) start releasing psychic energies, and the specific god(s) that people believe in will wake up and may begin influencing society or even the environment. Gods in this view are psychic vampires that require humans' psychic energies to stay awake and have power. Once humans stop believing in gods, then the gods (at least in those territories) go to sleep again (they become lethargic, as they cannot stay awake without humans' psychic energies), waiting to be awakened by humans' psychic energies again.
Potestas deorum ex hominibus: the gods exist and are awake 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, either 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 perhaps they just lose powers to influence the human realm but remain awake and healthy. 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.

Coming soon? Distinction between societal beliefs and individual beliefs (e.g. how powerful can a god or abstract concept be if it's believed just by one person?)

How powerful can a god be if it's believed by just one person?

1: No power at all. A god (or any other abstract concept) needs to be believed in by at least a certain number of people in order for it to have any effect. Gods and abstract concepts require communication in order to be effective.
2: Some power, but entirely limited to changing the perception of the believer. This idea, in and of itself, can have drastically diverse implications depending on your view regarding the relationship between "reality" and the mind.
2.1: Naive realism (the view that what we perceive is reality): very limited power, entirely limited to changing one's perception. Perception can alter behavior, beliefs and understanding of things, but it cannot change reality itself.
2.2: Idealism/panpsychism: the mind is the primary or only substance of reality. By changing your perception, you can change reality. To what extent and what effects it can cause exactly, is up to sub-views to decide, but at the very least, socially transmitted information (e.g. "children must be sacrificed to Tlaloc, otherwise there will be severe scarcity", or "if you're not a Muslim, Allah will torture you forever after you die", or "Kim Jong Un is a god-like figure that must be celebrated" etc.) drastically alters one's perception of "reality".
2.2.1: The only thing that changes is human behavior and thought. It doesn't change the "external reality" in a physical sense, although, in an idealist/panpsychist view, that is not super relevant, as the mind, and subjective/intersubjective experiences, are prioritized.
2.2.2: Thoughts can alter reality in a more "physical" sense. Humans' senses are greatly influenced by language, so by having certain ideas repeated since childhoold over and over again, sensorial perception (and therefore reality itself, in a deeper sense) is altered directly.
3: The power varies depending on one's mind. People who have higher psychic skills (which can, depending on the view, be innate, can be acquired through training, or both) can do more things by believing in their deity.
4: Gods have no power, regardless of how many people believe in them. It's just a sociological phenomenon.
4.1: Naive realism/nominalism: sociological phenomena, such as "ideas have consequences", do not imply that the ideas "exist" in any meaningful sense, or that they have any inherent power.
4.2: Idealism/realism: ideas (including gods) have their own existence, and therefore power, but only within the context of sociology. They have no power outside of influencing human behavior and thought. How much power that really is, that depends on your ontology.
4.2.x: similar to 2.2.x