The DT file has the wrong title, is inaccurate and incomplete, and is not credited to its writer (William Allingham); and, as you have spotted, has nothing whatever to do with the well-known set dance tune you're looking for. Corrections have long since been posted but have as usual not been incorporated into the database.

The midi file at Mudcat Midis is the right tune, though preternaturally slow (concertina.net is better). Stivell, by contrast, recorded it at a ridiculously fast pace, quite unsuitable for dancing. I see that "The Session" describes the tune as a hornpipe; it isn't, strictly speaking, though often played as if it were (or, even worse, as a reel). They now require unnecessary registration, it seems, so I won't bother with their site again. There are better-informed, more accurate sources.

"Faeries" is best avoided as a spelling when searching; you will mostly find adolescent, romantic, imaginary tosh associated with it. Mind you, "fairies", while more down-to-earth and sensible, will probably return a high proportion of irrelevant references to sexual disposition along with the expected folkloric stuff. What can you do?