As an American, I would say either construction is fine for almost any musical instrument. "I play the piano", "I play piano"; "I play the trumpet", "I play trumpet"; "I play the drums", "I play drums". As far as when to use them, I feel that using "the" is better as an independent statement: "I'm Brian, and I play the guitar." Speaking without the article is more common when the phrase is part of a bigger statement or in response to something: "I play guitar in my band", or, "Do you play an instrument?" "Yes, I play guitar." These rules are not 100%, however, as this is English, so feel free to say it with or without the "the"!

I agree with Brian but additionally I wanted to say I think you're overthinking literal translations. Also, unless you say "I" play guitar, "play guitar" on its own sounds like a command. As Brian said, the rules are pretty lenient here and "play guitar" "I play guitar" "to play the guitar" "to play guitar" are all technically correct sentences in their own right.