Well, your professor is a little bit wrong
Yes, Luna is the Latin name for the moon, but its
official name, as approved by the IAU is, wait for it... the Moon. With a capital 'M'. The name 'Luna' is commonly used in fiction, but that's just poetic licence. Anyway, why use the Latin? The Greek name, Selene, is quite nice too.
It's the same story with the sun. 'Sol' is used a lot in fiction, but the IAU approved name is, you guessed it, the Sun. With a capital 'S'.
Incidentally, 'The Milky Way' is also not the official name of the galaxy in which we live. Wanna take a guess? Yup, it's simply called the Galaxy. With a capital 'G'.
Not that it matters. If you say 'Sol' or 'Luna' or 'Milky Way' or even 'Terra' people know what you're talking about and there's nothing wrong with using a little poetic licence here and there! And who knows? By the year 3301 those sci-fi names may have been officially accepted (more likely if we've colonised other star systems by then).