"Songs (with words) can't really be said to have different names,"
In my experience the opposite is the case.
The older singers tended not to bother too much with titles, taking a prominent line of a song to identify it. It was the researchers who listed the songs under standard titles.
A Travelling woman gave us four/five verses of a song she called 'My Brother Built Me A Bancy Bower' - we chose not to correct her by telling her that the 'actual' title was 'Famous Flower of Serving Men' (Child 106).
My favourite story on this subject was of a collector friend who was recording songs from an elderly singer in North West Clare.
The singer had a large repertoire of quite rare songs which my friend was quite anxious to get down on tape, but the singer kept asking him 'Do you know 'The Old Armchair'?"
Finally my friend agreed to record the song, and the singer began:
"Knight William was sitting in his old armchair,
Lady Margaret was sitting on his knee" -
It turned out to be 'Fair Margaret and Sweet William', the only version of Child 69 to have been collected in Ireland, and extremely rare elsewhere.
Jim Carroll