Getting back to my posts of 2010, I'm starting to think that the tune played on the Chieftains Live! really *is* Eamonn a Chnoic, mistakenly identified as the very similar version of "Caitlin ni Triall."

They aren't identical, but they're clearly variants of the same tune.

The nice version of "Triall" played here on the whistle

https://www.youtube.com/watch?v=JW7Xz59Jzcw

is different but related.