"But why would a 'folk' singer bother to differentiate these categories in the first place"
Because they are different - why wouldn't they ?
The old crowd actually had their own categories for their songs - Walter Pardon called his folk songs "folk songs", Tom Lenihan called them "the old tradition"
Blind Traveller, Mary Delaney referred to hers as "my daddies songs" - she knew around a hundred, when we recorded her father he knew about six.
Mary used her title as a description of the type of song she sang, not where she'd got them from
Another Traveller, Mikeen McCarthy went further and described different ways of singing his songs - one way to sell his ballad sheets on the streets and in the fairs, another way to sing them in the pub and, most important to him, the "fireside songs" which were sung on the sites among friends and family

A good song is only a good song if you like it and it does
something for you
Nothing to do with ethnomusicology - it's my experience that, even if well past their prime the older singer brought something into their songs which most younger ones never managed - conviction - they sang their songs with a lifetime of familiarity behind them
If you can't learn from that you really are missing out
The success of the present Irish scene was built on the fact that there were enough of the older generation still around to learn from and enough sensible youngsters to realise that fact and sit down with them to listen to what they said and did
Jim Carroll