Maybe a musician could chime in here [:smile:] to correct me if I'm off base. But I don't think it is "tempo" that determines if music is best for waltz or rumba; it is the pattern of beats and stresses or rhythm. I think the music that OP refers to is usually played in 6/8-time -- which can sometimes be danced as if it were 3/4-time (especially if the stress is on the 1st and 4th beat). The result would usually be a Viennese waltz, not a slow waltz.
These ballads are sometimes performed in a way that sounds like the music might be 4/4-time (like the Willie Nelson version) but I can't see a rumba here -- maybe bolero at about 80 beats per minute (or some kind of slow or night club dance).