It's easy to assume that a word has some sort of underlying notion, and that an Italian might think of "largo" slightly differently because it also means "generous" or "free". However, in practice, words often develop distinct meanings that don't necessarily affect each other. For example, the notion of bar (a musical notation, a pole, an alcohol establishment, or the law exam) are quite separate even if they may have had some common ancestry.

I am thinking it's a question of filters. I first encountered 'despacio' in a piece of music. Now that I'm living in a spanish-speaking country, whenever I come up to that word, the first filter is that piece of music; the second filter is the actual meaning of the word.

That would be Italian, wouldn't it?

As a fellow musician, I've wondered the same.

Your comment brings up a good question. Would a spanish-speaker transliterate those terms into a musical context, or into a language context first, since the words themselves are pretty equivalent?

Most are suficiently different (allegro would be alegre in spanish) or unused (andante) to keep the musical context. But it's easy to grasp the musical context from the language one anyway (in the case of largo and lento).