For me a cover is a new recording/performance of a song originally recorded/performed by someone else. In my mind it should be a pretty faithful rendition of the original but it can vary to reflect the aesthetic of the person doing the cover. If it does vary it should still be readily identifiable as the original song.
For me, that is a performance. A cover is when someone adds something that wasn't there in the original performance. As you can probably guess, I don't try to recreate what was there originally, necessarily, note for note. (One of my dear friends once commented, after I had played my heart out on a song, "You know that isn't how it goes, right?") But different strokes for different folks.