Not the only case of the Traditional Ballad Index lumping different songs.
Lighter, did Lomax ever put Georgia Turner on record? I haven't checked through the Smithsonian, thought you might know.

I haven't heard Ashley-Foster or Darby-Tarleton, The Ashley-Watson version (recorded for Folkways ca. 1960) is not bad, but seems to be a cover of the Lomax-Turner version (although Ashley said he first learned it from his grandmother).

In "Rise Up Singing," A note to "House of the Rising Sun,"(chords given), says "coll. adap. & arr. with new words & new music by John A. Lomax, Alan Lomax & Georgia Turner;" "copyright 1941 & renewed 1968 Ludlow Music Inc. NY, NY. International copyright secured etc. etc., All rights reserved."
I wonder how the "new words and music" differ from that of their originally collected song. Also in 1941, it was recorded by the Almanac Singers, not listed among the covers, who include Joan Baez, Doc Watson & Son, Weavers, etc.

The Almanac version restores the verse 'fill his glasses' originally included by the Lomaxes in "Our Singing Country" in 1941; the Animals keep pretty well to it except they change the order a bit and substitute Dylan's last verse, also leaving out the 'fill his glasses' verse.
Most people seem to have followed the Lomax-Turner version melody and "Rise Up Singing" chords; Josh White providing an interesting variant.