Its name is the Moon

The name of Earth's moon is "the Moon". In particular, its name is not "Luna".

Don't take my word for it, of course. Just look it up in the dictionary.

But they call it "Luna" in a science fiction story I read

That's fiction. If the Moon is called "Luna" in a work of fiction, that just means that's its name within the setting of the story. In the novel 1984, England is called "Airstrip One", but this in no way means that this is really England's name.

Fiction authors are generally well aware of this distinction. The Moon is called "Luna" in Robert Heinlein's novel The Moon is a Harsh Mistress, but when not writing fiction, Heinlein always referred to it as "the Moon". See, for instance, his nonfiction work Spinoff.

But the English language doesn't work like that. There's no such thing as official words for anything. Whatever word English speakers use to describe something is its name.

No, that doesn't apply to proper names. Specific people, places, and things do have official names, which can't be changed by common consensus. If we all started calling you "Betsy", would that mean that your name is "Betsy"? Of course not.

Also, even if that were the case, nobody actually calls it "Luna" outside of fiction, except science fiction fans in informal settings. In formal nonfiction publications that have undergone editorial review, it's called "the Moon". Some examples:

Astronomers may call it "the Moon" when they're talking to the public, but they call it "Luna" in professional journal articles.

No they do not: example 1, example 2, example 3.

Who has the authority to decide the official name of the Moon?

The International Astronomical Union Working Group for Planetary System Nomenclature, an internationally recognized group of professional astronomers. And they say:

The Moon does, of course, have a name - the Moon. — IAU WGPSN Planetary Names FAQ

But "luna" is simply the Latin/Spanish/Italian word for "moon"

True but that doesn't matter, since I'm talking about the English name for the Moon. Other languages have other names. The Spanish word for Germany is "Alemania", but that doesn't mean the actual name of Germany is "Alemania" when you're speaking English.

But "lunar" means "related to the Moon"

True but that doesn't matter. "Fraternal" means related to brothers, but the word is still "brother", not "frater".

I don't care what's correct. I like how "Luna" sounds, and I'm going to use it, as an affectation

Fine. I don't care. Just don't tell anyone else that its name isn't "the Moon".

See also