Nowadays we label both their religion and its modern equivalant "Hellenism" or "Hellenic paganism", but that is a modern label coined in the 1640s and coming from the Ancient Greek Hellēnikós, “of or relating to Greece or Greeks”.

If they had a word for "their religion", we simply don't know it. But saying that the Ancient Greeks, or any civilization at the time, had a single "religion" is very innapropriate. There was no seperation between "Church" and state. You grew up worshipping the patron gods of your particular city or region and those of your family. Worship in that family hearth-centered cult came before worship in the state-sponsored public temples or other religious institutions. Again, different regions would worship different gods and there was no enforced orthodoxy there. Cults often cross-polinated and were imported from neighboring regions. To make matters even more complicated there were dozens of "mystery religions", which we would classify as Hellenic paganism but are themselves destinct religions that were limited to fractions of the population.

I have a copy of Jane Harrison's Prolegomena on the Study of Greek Religion but never got past the introduction.

I always worry that so much scholarship is "of their time" and so nowadays would be superseded, possibly just by the current biases.

Is she still thought of as reliable?