I don’t know any more than the page tells me, but I think “exact” means “steepest descent” or “direct descent” down the local energy gradient, whereas “inexact” means “go someplace down the energy gradient, but it doesn’t have to be in the steepest direction”. Always following the steepest path probably gets caught in (more) local minima more quickly / more often, whereas more global minima are preferred. (Here by more I mean in the more vs less relative sense, not the counting sense).
Quench I think means “set the MC temperature to zero”. The Boltzmann criterion has temperature as one of its inputs. High temperature = energy-increasing moves more likely to be accepted. Low temperature = energy-increasing moves less likely to be accepted. Zero temperature = energy increasing moves have a zero chance to be accepted. Running MC at quench means “only accept moves that improve energy”. This drives energy down rapidly (good) but also traps the simulation in a local minimum – normally bad, but irrelevant here, because the simultation is ending and “trapped” means nothing when it’s done anyway.