Though, for the sake of simplicity, human-shaped elements can also be referred to as ‘characters’ despite that being imprecise.
I must admit that my own understanding of ‘characters’ in stories is a bit…not great. Though I’m actively figuring out how they work.

Characters are also an expression of setup/payoff, though I’ll admit that I didn’t realize this at first.
Basically, ‘agency’ means that the character is capable of making choices within a story. This is to differentiate them from other elements which are mostly static unless acted upon (by characters).
Where does the setup/payoff come in? Well, a character is the setup and their agency is the payoff to that agency. Though this is the super abstract layer and mostly irrelevant to storytelling unless you want to be super pedantic.

This is a (possibly) temporary section until I figure it out for real.
Each character should have a core characterization which informs the actions they will later take. This should be, like, the main beliefs of a character. It’s super difficult for me to explain this right now.
Other aspects of the character will naturally stem from this ‘core’: traits, beliefs, etc. Though what the core is can vary as much as you want: childhood trauma? Adult trauma? First birthday? Best day ever? First kiss? It can be basically anything you want it to be, provided it can inform their actions later on.
It would be easiest to just make the core a set of beliefs that the character holds, plus some past experiences which explain their current behavior. My own ‘framework’ for this makes use of a few ‘memories’ which then inform the beliefs/actions of the characters. Though I’m not saying that my framework is the best (I don’t know that yet).
Though you could, in theory, arrive at a core backwards: come up with a list of traits first and then try to find a means of justifying them. You could also tie the core directly to the theme of the story, which would give the theme more weight.
This is also what motivation should be tied to. Speaking of…

A word thrown around a lot with characters is ‘motivation’ and while I do think it’s important, I think that people end up forgetting the ‘core’ in the process.
To sum it up, the motivation is directly related to the goals of the character within a story and, as such, is story-specific. It doesn’t actually inform the personality, but does shape it.
I think it’s safe to say that a character without a proper core is a bad character, though it’s hard to articulate exactly why.

This section is for additional character-related analysis that I’m putting together mostly on the fly.

I’ve heard characters be placed into these two broad categories before and, honestly, it’s completely worthless.
Well, maybe not completely but definitely mostly worthless. Similar to Wattsonian VS Doylist analysis.
The idea here is that ‘reactive’ characters are those who mostly react to the happenings around them and then act accordingly. ‘Proactive’ characters are those who are more proactive. They don’t react to the happenings around them as much as actively make them happen. When I heard this, it was stated with confidence that proactive characters are universally better.^A)
The issue is that it’s extremely difficult to determine whether a character is reactive or proactive, because you can interpret a lot of reactions to things as proactive actions. Alternatively, proactive characters could just be reacting to something else.
That said, this isn’t completely worthless as a writer might find it useful to see whether their characters mostly react to events or not. But I don’t think it’s good or bad either way, though a ‘reactive’ character could indicate a weak one.

This is one of my ‘crack’ theories and its usefulness is To Be Determined, as I’m in the process of trying to figure out some more character-related stuff.
A stable character is one who can be ripped from their context and placed into a new one without feeling out-of-place. An unstable character cannot.
This might be useful to determine whether a character actually has proper characterization: basically, if you can reliably predict how a character is going to act in any situation presented to them. If a character is not properly characterized, they’ll be dependent on the surrounding context to sustain them. This is also why, I think, some characters just feel ‘off’ in sequels, cross-overs or other such stuff, while other characters still feel like themselves no matter where you put them.

Another ‘crack’ categorization. I won’t elaborate on it too much, since I came up with it just now.
Tethered characters are those who are deeply tied to the story they belong to. They aren’t necessarily unstable, but ripping them from their context is undesirable anyhow. Untethered characters are those who can be placed into virtually any context.
To give examples: Indiana Jones can probably be placed into any context and he’ll mostly be fine. Meanwhile, the antagonist and protagonist of Daybreakers cannot be placed into any context, because their characterization is directly tied to the plot and theme of that film.
This may or may not be a useful means of classification and, in fact, I’m not really that sure about whether a ‘tethered stable character’ could exist.