her little Scotch song of Barbary Allen

We'd use the same locution today, to imply no more than that she was particularly associated with the song. Which might well be the case even if it had been written 200 years before.

One reason to think it might be at least somewhat older than its first occurrence in print is that the the earliest broadside says it's to be sung to the tune of Barbara Allen. Which is not included. Steve Gardham has a "nowt so queer as folk" attitude to that, but it does make sense if the song was already current and the point of the broadside was to appeal to people who knew the tune but needed some help getting the words straight. (And I can't think of any other interpretation that maks sense of that).