For what it's worth:

Here's the definition found in one online dictionary: a recording of a song previously recorded usually by another performer. This is also the meaning used in the jams and song circles I attend. A song one plays/sings is either an original or a cover.

Historically, some songs have been covered not so much out of admiration for the song but in an obvious attempt to supplant sales of the original. One spate of this familiar to many was the covering of R&B songs in the 50's by white performers. Another example I have seen was a series of covers of 50's-60's chart hits that clearly intended to be mistaken for the orignals. In this case, the artist's names were deceptively close to the original (I recall "Clyde King" doing Claude King's "Wolverton Mountain"), and the arrangements, instrumentation, and vocals were virtually identical to the original.

The types of covers described above generally elicited a certain amount of disgust from those familiar with the originals, and a degree of outrage at the rather blatant and calculated attempt to garner sales at the expense of the original artist. Conversely, many covers appeared to be based more in an attitude of "That's a great song, let's record it." These don't seem to generate contempt for the covering artist (though they sometimes generate pity if the cover is feeble).

I suspect that the emotion attached to the despised covers has made the very close copycat versions stand out in some folks' minds, to where they associate the word "cover" with only that specific type of cover. I believe a study of the historical use of the word will support the more general meaning found in the definition above.