But here’s the thing: while you may disagree that the two-line description I suggested coded “chaotic” to you, absolutely nothing in the description coded “lawful”.
Although I suspect this isn’t your intent, the response comes off a bit “hey, you play that monk, as long as you still choose to be lawful”.
Which goes back to the original point: I cannot play a chaotic monk, for whatever value of chaotic the DM may apply, regardless of whether I agree with it or not.
D&D "lawful" encompasses two different sorts of concerns -
adherence to custom/tradition/social role and
self-discipline. When we narrow our focus to "lawful good" then we see a further notion that self-disciplined action, that adheres to custom/tradition/social role, will also make people as a whole live safer, healthier, more peaceful lives.
This will all work for a certain conception of the paladin,
provided that everything else is in place (eg the world is an Arthurian one, in which upholding the customs of the land through one's self-disciplined actions coheres with upholding goodness and the divine plan). It's easy to make it come apart, though, if we posit a Robin Hood-ish world, or the world of some martial arts films, where an evil king/courtier etc has taken over, so that
adherence to social role will undermine the traditions that have tended to make the people live good lives, and will rather lead to the oppression of the people and the selfish enrichment of the "false" ruler. Is Robin Hood CG (because opposing the "false" ruler, by pursuing his own individual path by striking at tax collectors from his hidden forest redoubt) or LG (because loyal to King Richard, maintaining a "true" band of retainers - the merry men - while trying to restore the traditional order by deposing the "false" ruler)? The D&D rules for alignment offer us no help in answering this question.
In the context of the monk, similar sorts of breaking-down of alignment coherence can occur. And because the monk's self-discipline can easily drift in a more ascetic direction
and a more esoteric direction than the paladin's, it's easier to set up a world context in which
adherence to custom, tradition and social role and
self-discipline come into conflict. For instance, the lawful monk might have to reject certain sorts of food, the normal trappings of family life, etc, and also have to perform certain rituals or uphold certain taboos (as part of the esoteric aspects of their self-discipline) - which can lead to obvious conflicts with custom and so on, if the world is not set up in a certain way. We see this variation in martial arts films, where sometimes the monks are a slightly odd but very welcome part of the social order, while sometimes they are dangerous outsiders whom the mainstream authorities tolerate at best.
Again, the D&D alignment rules offer us no help in answering these questions.
A further source of conflict occurs in the context of martial/military orders who self-consciously follow a code that distinguishes them from, and elevates them above, the ordinary warrior. Romanticised/idealised versions of knights and samurai are examples. These characters are self-disciplined, members of an order and tradition, immensely conscious of social position, etc - and so are clearly lawful! (Says the author of the paladin and the original OA.) Yet they are also hard for ordinary rulers and generals to discipline, tend to act autonomously and on their own individual concerns/motives on the battlefield (eg calling out enemy warriors for single combat), etc - and so are clearly chaotic! (Says the author of the old White Dwarf article
"Dungeons and . . . Dragoons?")
This is yet a further matter on which the D&D alignment rules offer no assistance.
I don't regard it as a criticism of the alignment rules that they offer no assistance for social/political/moral circumstances that depart from a
very particular set of assumptions: those are just the assumptions that the game is based on. I do get a bit frustrated by proponents of D&D alignment who try and maintain that the rules have some sort of coherence or applicability beyond those assumptions, as if they could provide some universal framework for characterising personalities and political/moral commitments.