The DT tune presumably belongs to  REYNARDINE,  which is an American version.  The same tune is also linked to from  REYNARDINE 2, which is a transcription from the Roberts and Barrand record.  Unfortunately, no indication is given that this is not the appropriate tune for that version.  The mixolydian one would be the set that everybody learned from A.L. Lloyd, I expect.  I don't think that Lloyd ever said where he got it; it's obviously related to the Reynardine tunes in the Petrie collection, but has a very different flavour.  I rather suspect him of re-making it from one of them into a more "interesting" form.