This is a fascinating and important topic, and a lot to digest! There's a reason that it was apparent saved for advanced study in Epicurus' day. I'll weigh in here but I've got limited time so hopefully I can make a coherent contribution.
Discussion of the gods is inseparable from discussion of prolepsis. So to start, I think an excellent definition of prolepsis is "the faculty of pattern recognition." This compares to the faculty of the senses or of the feelings. An individual prolepsis is true in the same way that an individual sensation is true: it's what is perceived, but it doesn't necessarily correctly represent the external physical condition (think of the square/round tower example).
Next, there is the particular prolepsis, in this case that of the gods (in the same way as there are particular sensations, for example). Finally, as I understand it, a particular prolepsis is not fixed but can change over time. Think of the infant's inkling of justice v that of an adult. A recognition of a pattern evolves as more information relevant to the particular pattern becomes available.
So for me, the core, and potentially unsolvable, problem is to determine what is the most primitive version of the "god prolepsis." It's certainly not a white haired, bearded man in a toga or a shape shifting animal as in some cultures. These are additional pieces of information added to one's innate prolepsis, by the individual's culture, from birth. The question is "what pre-dates this?"