So tell me which songs from the collected tradition had not been rewritten by some person or persons before the version that was sung to the collector?

You can't tell me, because it is unknowable. There is no unspoilt well, from which pure traditional folk songs can be obtained, clean of any human intervention/alteration/polishing.

In this case we have songs where the source turns out to be a known person, in which case the record should be annotated as such for specialists and archivists, and the rest of us can go on singing them or not, as the fancy takes us.