You are looking at this with a mindset that recalls doctrine and dogma, a modern analogy that would not fit folk practices, particularly in the ancient world. Your ancient Greek would have conceived of his world as filled with spiritual forces that came to play in murky, poorly-defined ways. If your ancient Greek leaned in his/her belief in one direction or another, either Zeus or Poseidon could have been seen as responsible; and that responsibility could change from one moment to the next.

We see religious beliefs is strictly defined by written documents and a formal priesthood that describes in specific terms how the spiritual world conducts itself. For the illiterate folk, belief is much more fluid.

I've provided the following excerpt before, but it may help with how to approach ancient belief systems; documents that record "myths" must be considered with source criticism and an understanding of the context of their time. This is from my newly-e-published Introduction to Folklore:

Something also needs to be said here about myth. People use this term awkwardly. In a European context, myths tend to be the artificial constructs of ancient and Classical-era priests or literate people who sought to weave folk traditions into a comprehensive whole. The exercise often had political purposes, designed to provide diverse people with a single set of beliefs and stories. By reconciling similar traditions, the shared culture of these groups could be seen as more important than the differences, justifying the central rule of the king and his priests. Myth is also a way of organizing and reconciling folk traditions, which by their nature can be contradictory and highly localized. Myth tends, however, to make gods of supernatural beings, giving those powerful entities a status – for modern readers – similar to the Judeo-Christian-Islamic God, even when this comparison is not justified. Of course, it is also important to point out that myths were stories that were told – and then written down – and they were different from religion itself. Many myths were simply the shared cultural inheritance of a group of people.

In general, the word myth is best set aside when discussing more recent folk traditions, recognizing its proper status as a literary genre. Nonetheless, ancient documents recording myths can assist in understanding the history of various stories and beliefs. The authors of these texts were, after all, the first folklorists, and they were the only ones coming close to practicing the craft at the time.

Some folklorists carelessly use the term myth to denote those legends that deal with a fantastic, remote time. This primal era saw the creation of many familiar things such as day and night, fire, animals, people, mountains, and all other aspects of the present world. Folklorists properly refer to these stories as etiological legends explaining the origin of things. Sometimes, however, people interchange etiological legends with the word myth. The problem with this is that “myth” can imply something that is inherently wrong, linked to “primitive” superstitious beliefs. When the term “myth” is used for the folklore of existing cultures or for the traditions that were viable only a generation or more ago, it can take on an insulting, derogatory tone. It is best to reserve the word “myth” for ancient and Classical-era texts.