Well; technically you do have it in the wrong category insofar as you have used the Steeleye Span text, which is a modern collation and therefore not traditional; though the songs from which they made it were traditional, and English, not Irish; so your correspondent is wrong about that.

The tune is another matter, as is so often the case, and turns up in a number of variant forms attached to other songs as well. I wouldn't like to guess what country might have earliest title to the melody, but there doesn't seem any particular reason to think that that would be Ireland, though it has certainly been found there as well as in England.

As Masato says, the song was discussed in considerable detail in the thread he has linked to. I think that I identified the sources of the examples in the DT there, too.