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Humans already named objects within the solar system after Latin words and names. Earth would be called Terra, meaning “earth” or “land”. Many sci-fi stories did this already lol. Luna is already reserved to Earth’s Moon. The other “moons” in the solar system already have their names from Latin like Europa or Ganymede.
We already call satellites orbiting a planet “moon”, as in Europa is a moon of Jupiter. Or Phobos is a moon of Mars. But right now when we say moon, it usually means our moon.
You’re right that in the future we would have to use a generic term for all extraterrestrial objects once we start colonising space. Which is why I think in the future, “moon” will become the catch-all term for the non-Earth moon, while we will call our own as Luna instead.