G'day Gloredhel,The story sounds great (folk etymolgies, by their very nature, always do)... but, in my old (single-volume) Oxford Companion to Music, Percy Scholes remarks that the tune we now call Greensleeves is a typical example of the kind of tune being brought into Britain, at that time, by the Italians music masters introducing the newfangled violin (displacing the British Crowd/Crydd/&c).
Regards,
Bob Bolton