Paddy Tunney's set of the song has appeared on recordings and in print variously as The Green Fields of Canada and as The Green Fields of America. It appears under the former title in his book The Stone Fiddle: My Way to Traditional Song (Dublin: Gilbert Dalton, 1979, pp 156-7), as he got it (eventually) from his mother Brigid. The DT file Green Fields of Canada purports to be copied from Paddy's book, but is not an accurate transcription.

The DT file Green Fields of America (3) reckons to be "from singing Paddy Tunney" (sic) but is very different. Maybe he had two versions of the song.

Also in the DT is Green Fields of America (2), taken from one of those Ossian anthologies that credit no sources, and Green Fields of Amerikay, from the Sam Henry Collection, which, though on the same subject, is essentially a different song.

There are corrections to the first DT transcription mentioned in this Forum thread: The Green Fields of Canada, with some comments from John Moulden, a considerable scholar of Ulster music, which are pertinent to your question.

You could find most of that yourself via the onsite search engine here, of course. What you wouldn't find that way is these broadside editions of the song, printed in the mid 19th century. They are at the Bodleian Library Broadside Ballads website:

The green fields of America

You'll see how the song has changed over time. "Authenticity" is very subjective, and opinions differ widely. Andy Irvine is a Revival rather than traditional singer, so, fine though his recording of the song was, it's an arrangement of a traditional song rather than a version in its own right, if you see what I mean; at least at this stage. I'm not familiar with Mary Dillon's background. In both cases, though, you'll get a more accurate picture of the song as it has circulated in oral tradition if you go back to their sources (assuming they say who they were; if I remember correctly, Irvine recorded a shortened form of Paddy's set).