Many Monks are Ascetics, despite what form it takes, they seek a higher spiritual state through discipline or self-denial (and some also practice self-mortification). If the name gets changed, that is what I'd change it to.
Ascetic DOES work quite well and is more focused on what they are than Mystic (which IS a bit too magic leaning, hence my early dislike of it)
All the monks are based around self-discipline (self-denial and self-mortification are more optional elements) and that works well.
When you said Ascetic the only other term I thought of that would be similar was "Devotee"
Some people would note that Drunken Master is a bit off, but they're still about self-perfection and they don't actually get powered by drinking (thank god).