Not quite.

Stevenson's lyric was published posthumously; it pre-dates Miss Bean's long-forgotten effusion, which was written at the request of Greig's publishers, who seem, puzzlingly, to have been unaware of the tune's provenance, though they gave it Boulton's title. In her Songs of Scotland (Edinburgh: Mainstream, 1996) Wilma Paterson quotes from Mrs Stevenson's account:

'The writing of "Over the sea to Skye" grew out of a visit [in 1887, to the Stevensons at their house in Samoa] from one of the last of the old school of Scots Gentlewomen, Miss Ferrier, a granddaughter of Professor Wilson (Christopher North). Her singing was a great delight to my husband,who would beg for song after song, especially the Jacobite airs, which had to be repeated several times. The words to one of these seemed unworthy, so he made a new set of verses more in harmony with the plaintive tune.'

Stevenson's lyric, set to the tune, was found among his papers after his death in 1894. He too seems to have been unaware that it was a modern song. It was subsequently published, with the music, with the permission of the copyright holders.

Is there actually any evidence that the 'Cuachag nan Craobh' tune was ever used for waulking (not waulkin', please)? It was heard by Miss MacLeod as a rowing song ('iorram') and there doesn't seem on the face of it to be any reason to think that it was 'originally' used for waulking.