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potato comments on (Subjective Bayesianism vs. Frequentism) VS. Formalism - Less Wrong
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Yes, we can use "probability is degree of belief" but we have to be very careful about this sort of word play, because what that really means is that "probability models degree of belief".
Probability doesn't come from attempting to model something out in the world. It comes from attempting to find a measure of degree of belief that's consistent with certain desiderata, like "you shouldn't believe both a thing and its opposite." So the phrase "probability models degree of belief" is false.
You're riht, I mean to say "probability theory models theoretically optimal degree of belief updates, gven other degrees of belief". Or "probability theory models ideally rational degrees of belief."