I have even been using the term "traditional cover" if the song's origin is unknown, like The Parting Glass. I'd use titles like this: "Liam Clancy - The Parting Glass (Traditional Cover)". Is that also very inappropriate?

Joe - even for us non-native speakers, we would not think those songs as "covers". On the contrary, we would think of the "other performers" who did the song first as doing "covers", even if they were the first to do the song. We'd use titles like "Keith Whitley - When You Say Nothing at All (Don Schlitz Cover)". That's the definition of a "cover" to us young, non-native speaking people..