My main citation for the Primary Moon being known by Eorzeans as 'Menphina' is that Dalamud was known in-universe as
Menphina's loyal hound, which makes sense - Dalamud orbited the Primary Moon (as shown by the models of Hydaelyn and it's moons on the
Astrolabe), but like real life where Luna is the fancy 'traditional' name for Earth's moon but officially is simply called the 'Moon' by the IAU (and pretty much everyone on Earth), it's never really called that by Hydaelyn's inhabitants, including Eorzeans - instead the smallfolk simply call it the 'moon' or 'the Primary Moon' when Dalamud was still around.
If Zodiark really was the name for the Primary Moon, I think that would have ended up somewhere in Hydaelyn's culture, yet it is the Ascians' who first reveal Zodiark's existence, his name not even being uttered at all until the final stinger of 2.0's initial main scenario and absolutely no sign of his name anywhere in Hydaelyn.
So even if Zodiark literally
is the Primary Moon (an idea I don't buy myself), no one on Hydaelyn seems to know about that.