What Pepys writes of is "Mrs. Knipp...and *her* [my emphasis] little Scotch song of 'Barbary Allen.'" That may suggest a greater exclusivity to Knipps, and far less familiarity to Pepys, than "the."

Since Mrs. Knipp was a noted actress, it becomes more plausible that the song originated on the stage. OTOH, the diary has she and Pepys singing at "Lord Brouncker's," not in a performance as some writers seem to have assumed. Yet in a subsequent note to Pepys, the lady signed herself "Barbary Allen." This might suggest equally either a stage persona or mere coquettery.

I may have overstated the likelihood of a stage origin (not that it would make the song less "folk," just a little farther from the platonic ideal); but the case is annoyingly ambiguous.