It was "promenaid" when I did square dancing as a teenager. This is not quite as long ago as the days of the dancing masters, but does suggest it's been pronouced that way for several decades at least.

An undeucated guess: could some of these terms have come from French Canadian dances, and then into the UK via American square dances?

Strip the Willow has probably some connection with - well, stripping the bark off a willow twig. You'd pull off a bit one side, a bit the other side, and so on --- no, not that hand! Right to your partner, that's it ... now left to the next ... no, no, remember you're a man, dance with the ladies!
I should think stripping a willow twig was less complicated.

Marje