I keep thinking Olympian makes sense because it's a bit beyond Greco-Roman. There was a central philosophy to the stories, and there are many similar characters. Melqart is Herakles who is Hercules, and the first in that list was a Phoenician/Carthaginian god.
Phoenician, Anatolian, Greek, Italian (South more Greek, central more Etrurian) and Phoenician colonies in Spain and North Africa seemed to share the same religion.
While all these cultures owe their literacy to the Egyptians and Mesopotamians / Babylonian cultures I suspect that their religions and myths were more distant. Certainly some legends and gods were shared with these people, but not as many.
I find it lazy when people say the Romans copied the Greek faith because it wasn't quite the same thing. Certainly the Aeneid was an attempt to make a Roman Odyssey but the south of Italy was heavily populated by Greeks, but the Romans had unique myths, the Italian peoples certainly did, and the two-faced god Janus is distinctly Italian which no one seems to remember.
I get the impression this religion has been poorly defined, whereas the Scandinavian and Nordic religion has been rather clearly defined in recent years and it's links to Germanic and Slavic paganic faiths are well known and understood. I'd also be interested to here where these faiths intersect with 'mediterranean' paganism
When I try to find a generalized version of this 'Olympian' or Mediterranean' faith I get Hellenic neopaganism. Admittedly when one looks up the Nordic variety one finds Odinistic neopaganism. I'm interested in the study and definition of the old paganism, not how it is expressed and ritualized today.
You're actually hinting at several (relatively) distinct phenomena here. I think one of the 'rubrics' you're looking for here is Indo-European religion/mythology/language. As the name suggests, this unites a wide variety of mythologies/languages etc. across Europe and India as having descended from a common ancestor.
Whereas the former similarities may be considered as due to "common descent," there was another phenomenon of diffusion. Here, there was probably more direct borrowing (of, say, Mesopotamian or Egyptian culture entering into 'Greek' culture). There's also the issue of things like, say, earlier Italic/Roman culture having an Etruscan substratum. Even though - to pick one thing - the Etruscan language comes from a totally different language family than Latin does.
Interpretatio graeca and Interpretatio romana might also be the name(s) of something you're looking for.