Q has already told us that the (uncredited) DT file is the Lomax text. I now remember that I said much the same (though in my case it was only an educated guess) in the other thread when somebody posted the text there, too. It appears so far that the only sources to call the song "Billy Barlow" were (Ruth Crawford) Seeger; who got the song from Lomax; Pete Seeger, who will have got it from RCS, presumably; and Silber, who probably also did. Looks as if it's really just one single variant in which Barlow features, though one much put about in print and on record. I can't imagine why anyone might think for a moment that it isn't a "Wren" variant; though according to the TBI there have been people who did.