But your original post said a good character shouldn't rely on other people's reactions. So we can't use Ahab's influence over his crew, right? And we see Atticus Finch and Tyrion Lannister interacting with other characters as well. I think I'll let this go, now. I understand what charisma is and what charismatic characters are like - I just disagreed about the "shouldn't rely on other people's reactions" part.
I agree with @BayView A character is other people's reaction to them - even if they are only conversing with the reader. Otherwise you just have a plank of wood.
Right? Unless a character is in some kind of soliloquy setting it'll be hard to tell anything about her without other characters passing through her wake. Charisma essentially reduces to the ability to lead and influence people, which can't happen unless there are other people around to be influenced and led.
I'm writing a charismatic character right now and it's semi-difficult cause he's kinda unconventionally charismatic – he can be blunt, rude, un-pc and obnoxious but … people are drawn to him. They like his frankness so they find themselves … instead of interrupting listening, stepping closer instead of moving away. He changes peoples normal reactions to the situation. He makes them think a bit more instead of just react. And they like him for doing that. He sparks something in them. He also has a very contagious laugh. I'm for relying - a bit - on other character's reactions but shake it up ... maybe it's not always what you figure. Instead of insta-like maybe just an insta-shake up.
Charisma is confidence, wit, charm and attraction. Have your character be fearless in the way they talk, unafraid to offend but interesting to listen to. Have your character be funny or amusing, but probably not in a self-deprecating way. Have other characters either love or loathe spending time with this character. Your reader's empathy will pick up on that and mirror it.