Of course it depends upon how nitpicky one cares to be, but I think it’s useful to limit “song” to numbers that are sung. For example, I like Irish songs, which have words and are sung. I don’t nearly so much like Irish tunes, typically jigs and reels which in my experience don’t have words. If someone asks my opinion of Irish music, it makes quite a difference whether we’re talking about Irish songs (yay!) or Irish tunes (ho hum).
Musical pieces that never had words can be called tunes, instrumentals, numbers, pieces, and probably a few other terms that aren’t leaping to mind – I don’t see a problem with NOT calling them “songs.” Non-sung versions of songs blur the lines a bit. Perhaps they’re “playing our song,” but no one’s singing – now what? Whether to call it a “song” or a “tune” may depend on the particular situation or other nuances. Then there are pieces that started out as strictly instrumental, but later had words written for them, such as “Sleigh Ride.” So now it is, or can be, a song.
Language use being what it is, a lot of folks will say “song” even if there are no lyrics to sing. I’d prefer they didn’t, but they don’t ask me.
ETA: You, dear readers, can join the campaign for increased clarity in speech by saving the term “song” for actual songs.