You can be pretty sure that's 'folklore about folklore'; an 'old wives tale' -or father's- if you prefer. People like to be able to explain things so that they make more sense to them or to others, and it's very common to find songs developing 'back-stories', often linking them to particular times, places or individuals. It can give a personal resonance -an enhanced sense of cultural ownership if you like- but that can also lead to a degree of protectiveness when two mutually contradictory stories clash. 'Are you calling my old Grandmother a liar?' You know the sort of thing.

Although this playground version of the centuries-old 'Cruel Mother' story may have Irish roots, it's found all over Britain as well, and it's pretty unlikely that we will ever know where or when it was made into the 'Woman in the Wood' / 'Old Mother Lee' form. You'll find other localized 'explanations' in some of the other discussions here of this song and its many relatives; see the list of links at the top of the page.