I'm sure that if Jay Ungar says that he wrote the tune, that he indeed did write it.

There will be tunes that influenced it and tunes that may sound a little like it but that does not mean that we should doubt his word. After all there are only so many combinations of notes that it is indeed possible that almost identical compositions are possible from two or more unrelated sources.

The small differences in tunes are lost on most people anyway. My sister-in-law thinks that every tune I play is 'Captain Pugwash', be hornpipe ,jig or reel. It could be the way I play them I suppose?