"Lord of the Dance by Sydney Carter is a re-write on a traditional folk hymn called "Simple Gifts" "

I wouldn't call it a "rewrite" - it's a new song, using the tune of the Shaker song. Shakers believed in dancing as a form of worship, which is the link.

"I adapted the melody" wrote Sydney Carter in hi scollection "Greenprint for Song". "I could have written another for the words (some people have), but this was so appropriate that it seemed a waste of time to do so. Also I wanted to salute the Shakers."

If variants make a folk song, Sydney Carter's have a flying start, since he believed in changing his songs, tune and words, as he felt like it. (Of Lord of the Dance he wrote "Sometimes for a change I sing the whole song in the present tense.'I dance in the morning when the world is begun...' It's worth a try.!)