Other common names to be noted are Young Jemmy Grove and Sweet William ~~ and you omit to note, Suasan, that he was Sir John Graeme of the West Country, which location doesn't sound very Scottish, altho I acknowledge that the name does. As for it's being thought of early on as a Scotch song; that is how Pepys described it in what is generally taken as the first ref to it, in his record of having heard the actress Mrs Knipp sing it:
--- A diary entry by Samuel Pepys on January 2, 1666 contains the earliest extant reference to the song. In it, he recalls the fun and games at a New Years party:[3]
    "...but above all, my dear Mrs Knipp with whom I sang; and in perfect pleasure I was to hear her sing, and especially her little Scotch song of Barbary Allen."--- wikipedia

I have signed like this

~M~

for years. I got there first and URA copycat!