Amos, you are not confused; they are indeed two different songs.
As far as I Ride An Old Paint, I usually sing:
Old Bill Jones had two daughters and a song, One went to college and the other went wrong.
While I don't think it's literally word-for-word, I believe it's quite similar to what I found in Carl Sandburg's American Songbag. I don't have the book handy at this moment, so I can't speak with absolute certainty.
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