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It's the "Carrying On" that is important; isn't that what Tradition is about?

"Carrying on" with variations. Obviously.

So, without variations, with no known singers till the '50s, where does "The Coal-owner" fit in?

The point, if there is one, is that in practice the category of "folk" or even "traditional" song can expand or contract according to the preferences (I almost said "whim") of the speaker or writer.