I don't think the meanings are that precise. I think the words are roughly synonyms.
I don't think the definitions you are using are quite accurate. I don't think the words mean "how accurately we judge our own abilities."
To me, "modest" is the opposite of "proud." Proud refer to "showing off." They have echoes of an older, literal meaning of "swollen" or "puffed up," like a rooster's comb or a peacock's feathers. There is even an expression, "as proud as a peacock." "Modest" doesn't mean that someone underestimates their own abilities, achievements, or wealth. It means that they _hide_ them. For example, a proud person might wear a class ring to show the world that he graduated from a famous university. A modest person might not.
To me, "humble" is the opposite of "arrogant." "Humble" means meek, submissive, acting as an inferior, obeying orders from others. "Arrogant" means bossy, overbearing, acting as a superior, giving orders to others.
"Modest" and "proud" emphasize appearance: what is shown.
"Humble" and "arrogant" emphasize behavior toward others.