When it comes to other people's characters feeling 'real' and 3-dimensional, I think that doesn't really happen until I get a chance to see them in different emotional states: sad, angry, happy, whatever their default state is, etc..
This doesn't have to be deep ('angry' can be something as simple as watching the character interact with a coworker they don't like), it just has to make sense. All their emotional states should blend together in a logical way, where you can point out the connecting threads. Like, 'ah, this situation makes them angry because it reminds them of that situation, in which they were too afraid to do anything, and their pride won't let them forget it'. When you have enough information to rationalize a character's behavior in that way, I think that's the point at which they become well-developed.
Sometimes writers try to give that information in ways that are kind of low-effort, though, and I think that's where things get confusing for fans. To give a common example, the 'positive sunshine sweetheart MC', a pretty popular character archetype.
Most writers try to round them out by showing occasions where they become genuinely angry or sad, or even mean, in direct opposition to their default state. Which isn't a bad strategy on paper.
What makes it bad is when they don't try hard enough to make it make sense: like, only a single person can make them mad (not a single type of person, literally just one character) or the lesson they 'learn' from becoming angry is to just never be angry again, instead of how to understand and manage a normal human emotion.
Or maybe the signal to the audience that the character has left their default state is to say 'I'm mad' and then proceed to act exactly the same. ^^; These things are fine when played for laughs, but when played straight they tend to cause problems, and often make the character seem more 1-dimensional, not less. Because they're simply not realistic.
In my own work, however, characters become 'real' when they start to speak for themselves. ^^ As in, instead of me carefully crafting what they say to try and teach the reader what kind of person they are, the character's lines start to pop into my head fully formed. They begin to teach me about themselves as I write, all their likes and dislikes and idiosyncrasies...at that point, it's up to me to just pay attention to what they're saying for future reference, and edit it so that the story stays on track. They become kind of 'autonomous', in a sense.
I'm gonna be honest: I think most of those arguments boil down to people simply mixing up definitions, and deciding that anything that sounds negative is an apt description for a character they don't like (for example, deciding a female who has a bigger role in the story than they prefer is a 'Mary Sue').
In this case, I think a lot of people mix up 'flat' with 'static'. Some characters can be incredibly complex and nuanced and interesting, and just stay exactly the way they are throughout the story (you learn more about them, but they remain the same person).
But because people are used to 'character development arcs' and dynamism in general, their first instinct is that something is wrong with a major character who doesn't change. So some may start calling them 'flat' or 'underdeveloped' compared to the more dynamic characters, even when it's not necessarily true.