I think the word "cover" originally meant doing an exact copy of an arrangement, but the definition has drifted in common usage to where it now means doing a song that someone else has previously done, regardless of arrangement. I have seen it used of traditional music when talking about a specific version of a song that has been done by many people, regardless of arrangement.

Strange how words drift in meaning. The word "gig" now often means any paid engagement in any field of endeavor, including a full time job. When I first heard the word, it always meant a specific musical booking.

John