How did "poll" ("top" or "head") semantically extend to "cut someone's hair"?

Compare bald head, and bald. I suppose you can synchronically form to boldly bald someone where no man has balded before.

I guess to poll may be similar. So far my thoughts a priori. This mostly disagrees with the quote that clearly distinguishes shaving from to poll, unless they distinguished only shaving the mustache specifically. If the quote is far removed from the original word (certainly not of biblical age), it might be indecisive.

The Ayto entry may have followed the written evidence, without giving too much consideration to semantics because it's actually not too difficult.

For analogy, consider a) German Schopf. As far as I understand it, that's lexically hair, a tuft or tail, but figuratively the whole head, metaphorically too (e.g. you have to grab the luck by the Schopf). b) Cp. Frisur "hair-do", frisieren "to do sb. hair", Friseur "coiffeur". Admittedly that's rather abstract, still, not exactly top-hair, but French frisure at least may mean "curl" (seems to come from Germanic but further origin uncertain, as are a surprising number of hairy words).

So (a) and (b) together allow to infer (head ~ top hair) > (head dresser ~ barbar) > (to head ~ to do hair) and all the same for poll.

By the way, I guess we may cp. shape and scape.