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.