There are three tunes in Johnson's SMM associated with Auld Lang syne, the first two are quite different to each other and to the one normally used now which appears with the title 'O can ye labor lea, young man' (sic) and was originally called 'I fee'd a lad in Michaelmas'.
I suspect like Peace, that Eddi Reader sang a rather free melody of her own which I personally didn't like, before going into the 'I fee'd a lad..' melody which was familiar to the audience.
I have just heard the Tannahill Weavers' version of 'The Braes of Balquidder' and the melody they sing to the verse is dire! I suspect they fudged it to make it sound more 'folksie'. They also mangle Tannahill's original words. Very sad.