Because my point that you bolded is specific to my approach to defining a character as a person, I don't think it makes sense to apply it to the wider discourse. Everyone is going to have their own opinion on what defines a character as a person.
From my perspective, having the quality of being "a great swordsman" does not require connecting with every attack roll, just as having the quality of being "knowledgeable" doesn't require knowing every obscure fact. The game mechanics are simply used to determine which specific attacks hit and which specific obscure facts are known. By contrast, I see having the quality of being "cool under pressure" as requiring a fundamentally higher level of consistency that reliance on random resolution can't provide.
In other words, I see having the quality of being "cool under pressure" as implying a degree of dependability that is contradicted by the uncertainty involved when relying on a game mechanic for resolution. So if there's sufficient uncertainty that you need a game mechanic to determine whether a character stays cool in a particular situation, then, from my perspective, that character isn't cool under that type of pressure.
I understand the parallel that you see between hitting in combat and staying cool under pressure, and I respect it, even though they don't look parallel to me based on how I approach defining a character as a person. Do you understand the difference I see between hitting in combat and staying cool under pressure, even though they look parallel to you based on how you approach defining a character as a person?
If so, great! We've reached mutual understanding. If not, and you have follow-up questions to try to better understand the difference I see, I'll do my best to explain. If you don't understand the difference I see and don't want to continue discussing it, that's also fine with me, I just ask that you accept that the difference is real to me and informs my preferences with regards to game mechanics.
I get that you are trying to make a distinction between these two things, but I am having difficulty seeing a difference. If a distinction exists, at most it feels like a distinction without a difference for me. The degree of dependability for being "cool under pressure" is, again, a factor of my overall degree of long-term successes with such tests, much the same for me as being a knowledgeable sage, a skilled swordsman, and the like. I don't expect that I will succeed every time, but if it's important enough to my character's character, then should I not be creating that character intentionally so they will have a higher success rate at it in whatever game system we are playing? Earlier I even showed it could be done with Cortex Prime's
Tales of Xadia. It's not 100 percent dependable, but it's not so random or unreliable as one might imagine.
One can certainly build your character to more reliably succeed than others, but it's no guarantee nor do I think that it should be guaranteed. Otherwise, it's basically creating a character whose "coolness under pressure" is never actually tested with any real meaningful significance, especially not if I decide my character's success and failure at being "cool under pressure" entirely by personal fiat. And if I have full authority or dependability for deciding when my character is "cool under pressure," then why do I lack such fiat of authorship about other qualities of my character's sense of character, such as their sagacity or athleticism? Is this not also something that I might deem as "dependable" for the sake of my character concept or their sense of self? I don't really see the difference here. It may be that my understanding of a character or even my sense of self is a little more holistic than yours. A character/person is not just the ghost in the shell, but the whole person/character in their entirety.
So is their "cool under pressure" characteristic really all that significant to the fiction if it's not put to the rigors of the test? Not from my own perspective. It's a bit of a Czege Principle violation going on. The player is essentially controlling both the adversity and its resolution for being "cool under pressure." That doesn't feel all that fulfilling or fun to me if being "cool under pressure" is something that I feel is important to the character of Tom the Fighter but it's never actively challenged. The protectionism of the trait, ironically, makes that quality of my character feel insignificant and orthogonal to their character. It's as if Tom the Fighter never got to fight and put their swordsmanship to the test or if I simply declared auto-success for their swordsmanship. It's as if a student walks into the classroom for their exam that their teacher prepared, declares that they passed, and then walks out.
If we were playing D&D, then being "cool under pressure" may require a Wisdom check or Will save, and I may want to build Tom the Fighter to have a higher Wisdom or way to reliably make Will saves. Herein, the issue of magic in this regard feels like a red herring to me. My character of Tom the Fighter exists in a world of magic. Being "cool under pressure" is not simply a mundane matter but also about being "cool under pressure" in a world/genre where magic exists. What being "cool under pressure" means in such a millieu must take magic into account for such a character concept. If Tom the Fighter is only "cool under pressure" when not dealing with magic, but magic involves the primary means by which his coolness under pressure is tested or overcome, then I don't think that I would feel like I am a playing a character who is "cool under pressure" when I am persistently failing to magic. That's why when playing D&D, I would likely want Tom the Fighter to have good and reliable way to make Will saves or Wisdom checks to represent their ability to stay cool under pressure with magic or otherwise when called for by the GM's challenges.