Sounds like a simple question at first.
The good news? I'm not going to go into music theory.
Nevertheless, the question is serious. What exactly is a song? Pick up a score, a set of tabs, but it's not a song. Sure, I could learn to read it, and to some extent then, "hear" the music in my head, but it's not the song.
Compare the following:
Are they the same song? Why not? Sure, the arrangement is different, the singers are different, and in this case, the styles are radically different, yet there is a commonality there that any normal person would say that either performance is of the song "Africa". We'd talk about them being different versions/arrangements/etc., but then, the above example is only different in degree from the case of live vs studio versions of songs, which often are played with different sounds and arrangements, even if it is the same band, same songwriter, and instrumentalists.
For the record, and with profound appreciation for Neal Pert, I tend to prefer live Rush to "studio" Rush, with the versions from a Show of Hands especially being preferred over the album versions of their older tracks.
The point is, as I close this out, that a song isn't a particular set of notes played on a specific instrument, sung in a certain way, by a specific person, or a piece of sheet music with notes and tabs. It is instead something expressed in some way by all of those, that nevertheless, while immaterial, is true, and exists.