I wonder if part of the problem with using alignment in the same terminology for both individual PC/NPCs and the extra-planar creatures is that a person who likes freedom and hates rules might be chaotic, but they aren't chaotic in the bad guy end of the world sense of the early Elric books, for example. A person who is evil in that they take what they need or are one of the evil protagonists in another thread going on seem very different from a demonic/devilish being whose goal is to actively cause as much pain as possible.
I'm wondering if something akin to the following would be useful?
LAWFUL - Following ones guiding code and working against CHAOS is the most important thing ("extraplanar lawful beings")
Lawful -Tries to always follow their guiding principals, but may have other over-riding concerns ("lawful")
lawful - Likely follows the rules unless they're standing in the way, but doesn't angst over it ("lawful tendencies")
chaotic - Chafes against the rules but doesn't go out of their way to break them just for the sake of doing so (unless they're annoying) or trying to sow randomness ("chaotic tendencies")
Chaotic - Flouts the rules and doesn't follow a personal code. ("chaotic")
CHAOTIC - Overthrowing the order - both local and universal - is the most important thing. ("extraplanar chaotic beings")
and
GOOD - Helping others, and avoiding harming others if possible is the most important thing. ("extraplanar good beings")
Good - Tries to help others when possible but sometimes knows that sacrifices must be made and may have other goals and purposes ("good")
good - Generally dislikes harming others and has an active conscience ("good tendencies")
evil - Doesn't particularly mind harming others and does so without hesitation when it serves their purposes but may have a group they look out for ("evil tendencies")
Evil - Enjoys harming others and causing pain, looking out for oneself is worth hurting others. ("evil")
EVIL - Causing pain, sewing despair, bringing woe, and actively overturning the GOOD are the most important things. ("extraplanar evil beings")
For something in the ballpark of those, should it be impossible to be either LAWFUL or CHAOTIC and also GOOD, since the prime goals would conflict? Would it be hard to be even Chaotic GOOD (as opposed to chaotic GOOD or Chaotic Good)? Would classic paladins be one of LAWFUL Good, Lawful GOOD, or maybe Lawful Good? Is there enough pain that could be sewn by overthrowing order that CHAOTIC EVIL is still a thing? Is there enough pain that can be sewn working within the system that LAWFUL EVIL is still a thing? Do the CHAOTIC EVIL seek the end of all things, so that many classic D&D Demons are actually Chaotic, while the things from the far realms are CHAOTIC?
Are there stories of the fae where they are bound by their oaths and promises, but generally random? How does one fit that in to an alignment scheme? How do needing to be truthful and needing to keep oaths overlap with alignment? How does that transfer to devils?
Is the chaos of the far realms different than the chaos of the devils? Does it lead to entropy and the annihilation of all things?
How do law and order differ? Does one of them lead to stasis and rigidity? Is Balance Lawful, but not LAWFUL?