I'm not sure that there's a "right" way anymore or that the original melody is really even known for sure.
Here's one with the original words of the poem and what "claims" to be the original melody. It's certainly different and quite enjoyable. Whether or not it's the original melody, I'm not sure:
https://www.youtube.com/watch?v=86_tlA9maA0I'd be inclined to believe that the Celtic Woman arrangement (pre-2016-HfC-tour) and the one in the video above are more like the original than is commonly sung in North America. From what I understand, it originated as a melancholy song, but one with a faster tempo than Lisa Lambe (and Eabha's) version. I believe the poem was originally put to the music of Sir Alexander Don's "Strathspey" (a type of dance).
It got adjusted over time for purposes of celebration and my own opinion is that the ascending notes in a major key (as opposed to the "traditional" descending notes) evolved for that purpose.