The Greeks viewed their gods as more human, very human in fact. Just one big dysfunctional family with the power to cause thousands of deaths, curse people, etc., no big deal. The Roman view of the gods was more complex, inherited partially from the Etruscans, and the ancient Latins, part of the reason I hate when people say that Roman religion was basically Greek. The Romans had the concept of numina, that every living thing and inanimate object had its own spirit, had its own "want". Stones want to stub your toe or hit hard, the river wants to drown you, etc. Of course it's more complicated than that but you get the gist, especially when you consider all their odd gods and rituals. Gods of crossroads, gods of opening and closing doors, gods of keeping the jars unbroken, all kinds of bizarre stuff, little superstitions and rituals.
While I agree with your description of numina and the way it interacted with an individual's daemon, I don't think the ancient Greeks viewed their deities as very human at all. Post-Socratic thought in Greece began to manifest a nearly anti-theistic sentiment in that the gods were to be feared when venerated. A family's personal daimon on the other hand was a household deity at could always be trusted. As early as Homer there is differentiation between gods (Theoi) and lesser deities (daimon)
Some say these beliefs stem from the animistic faith of the Dorians which was losing moral authority due to the extreme weather conditions such as the start of a Little Ice Age, the Hera-3 event, the mega drought that 'rendered the entire Mittani Kingdom dry' and a few more. When migrations would have started the Dorians would have turned to heroic ancestor worship which they implemented when they settled among the ruins of the Myceanaen cities. Eventually the fusion of the proto-Greek Mycenaen pantheon with the Dorian ancestral shines gave birth to the Greek religion, cutting out many of the elemental Mycenaen deities such as Proteus (Poroteu) by rendering them as villainous Titans or making them ineffectual in myths and legends such as Talos (Tawolo) whom recent finds indicate as a Minoan and Mycenaen God of minerals and the earth.