"It's also worth recognising that music hall songs, swing, boogie-woogie, ragtime, rock and roll and the rest were adopted by people"
They were not 'adopted' by the people - they may have been listened to and sung by them, quite often by the same people who sang folk songs, but they remained recognised as products of the music hall and the popular music composers and they tended to remain unchanged and unadapted. On the other hand, folk songs, whatever their origins, were taken over by the communities where they thrived and adapted out of all recognition. It would appear that their origins were considered unimportant enough to have been forgotten almost immediately on their being acquired (can provide enough evidence for this if required).
".....and even if altered contained the signature of the originator"
No they didn't, otherwise we might have some idea who wrote them - or maybe you have some idea who wrote Barbara Allen, or The Unfortunate Rake, or even what part of these islands he or she hailed from.
There is plenty of evidence that some folk songs were the original comositions of a number of people - all unknown.
The definition of a 'folk song' has nothing to do with who composed it or even in what form it was originally written - it is the process it underwent that provides its identity.
Jim Carroll