My understanding of the original meaning of the term "cover" in the present context was that it was a recording of a song other than the original that attempted to copy the original arrangement exactly. Several of the "homogenised" RnR/RnB versions of the late 50s pretty much fell in that category. Also, over here in the UK we would quite often get a cover by a local artist of a song that had been successful in the US. These copied the original arrangement as nearly as they could though most of the time, they didn't really get it quite right but became hits because of delays in releasing US recordings over here.

More recently, like it or not, the term "cover" has come to refer to any recording of a song, professional or amateur, that was originally recorded by someone else regardless of how like the original it is. I think it's a misnomer but that's language for you. If the meaning of a word shifts there is very little that individuals can do about it.