Strictly and scientifically speaking (as determined by the International Astronomical Union which is in charge of regulating the naming of celestial bodies), its English name is "the Moon" (note the capitalization and inclusion of "the" in the quotes), and the same goes for "the Sun". Luna (and Sol) are the Latin names for them, but the official modern names are just the Moon and the Sun. The Latin versions are almost exclusively used in fiction to invoke a sense of "scienciness" or advanced civilization, or in poetry just to be... poetic... but nobody uses them in the scientific community and literature, just like no scientist seriously references Earth as Gaia or Terra. Other languages may use something closer to the Latin (like Spanish la luna), but in those languages it doesn't sound foreign or special, it's just the way they happen to say "the Moon" in the same way our Spaniard might call a dog a perro. When it comes to Menphina I do however concur that very few citizens would refer to it by name, and just simply say "the moon" as suggested.
(Fun fact: professional astronomers don't call our galaxy the Milky Way in journals either, it's just "the Galaxy")
As far as our attempts to determining the latitude of Eorzea on Hydælyn, drawing conclusions based on that screencap do of course depend on our European cultural notion of north as being "up": our Australian friends may want to have a word with us... Of course, with the developers being Japanese, it's again not a far stretch to assume this is the convention being used, especially given the orientation of the in-game compass, no matter how much calling south "up" would be just as valid a viewpoint.
(For the record, I don't think we can actually glean with any confidence anything about the axial tilt of Hydælyn's rotation relative to its orbit based on current evidence, including the light bloom. The planet isn't shown to rotate in the trailer so it could just be camera perspective. Hell, we could attribute the apparent lack of in-game seasonal weather effects as evidence of a 0° tilt, though we run into problems with frozen-time paradoxes as once explained by Ferne in relation to main story events needing to be pegged down to a given year, regardless of when a particular player completes the content.)
This ended up way longer and more nuanced than I originally intended... but that's what I studied in grad school, so... @.@