Versnell Coping with the Gods: 54

Even if we are ready to dismiss the idea that knowing the name of a god implies having power over that god, this should not make us forget that knowing the name is an absolute condition for getting access to the divine source of power.

So yes invoking the names are important. But also the context, the local tradition, the personal relationship with the god, matter when invoking it. An Athenian merchant invoking Athena Polias in a law court is very different from a Corinthian invoking a different epithet of Athena. I can dig around for more specific examples later but etic navigation of the Greek world of Gods is a minefield.

As an addition to the question: Did the gods' epithets matter when using them for "swears"?

By speeches do you mean plays (sophocles etc.) or actual legal speeches (lysias) or poems (homer)? I could probably give you an answer for two of the three...

In a similar vein, you often have characters in plays or epics (or even historical figures in speeches) evoke "God," without specifying which god. In a pre-Christianized world, would this refer to a specific god or simply an abstract concept of a supernatural force?

I know the Romans would use "Mehercule" as a swear, By Hercules!, which was in part inspired by the Greeks, if I remember correctly.

I have attached an interesting article that goes a lot more in depth about this: http://www.shattercolors.com/nonfiction/baldwin_swearing.htm and talks about a lot of Greek examples.