If someone asked for the Russian words of the Lakes of Pontchartrain, wouldn't most people assume that meant the words in Russian?

Admittedly that first link An Pluiméir Ceolmhar gave does appear to rather bizarrely imply that it is Irish in origin (rather than Irish by adoption, which is true of just about any good song.) But then that happens all the time.

And I still am curious to know the Irish for alligator.

Normally when people ask for this song don't you find they tend to say "The one about the alligators..."