Hi Richie,
This is just off the top of my head and I will check in more detail later, but the 2 distinct strains of BA are this one and the Reading/Scarlet town strains. I would guess that the Sir John Graeme strain is likely the stage ballad referred to by Pepys in the late 17thc. The Scots language doesn't necessarily mean the ballad came from Scotland. Stage Scotch songs were very popular in the 17thc just as stage Irish songs were popular in later centuries. The 2 strains appear both to have long pedigrees in Britain.

The name Graeme appears in several border ballads and is very likely just generic. Even if you could relate it to the Graemes there were very probably many Sir Johns as I found when I was trying to trace another Scots family mentioned in ballads.

You can easily get 'Martinmas' by Googling.