Gibb,
It makes no mention of a capstan but clearly states 'add new vigor to their exertions, as the heavy yards and sails are mounting upwards....' Of course this might be done using a windlass.

BTW what's your earliest description of task and actual named shanty in references to British ships, excepting capstan and 'Cheerly Man'?
Regardless of where they originated I still feel strongly they were well established on any reasonable scale on American ships before it became standard practice on British ships. I know there's this 'mid-Atlantic' idea that seamen and ships were all one, but all the main impetus and input seems to come from American customs.