I think we almost agree already ;-)

Would you agree that "P(X)" you're describing is more like "some person's answer when asked question X" than "probability of X"?

The main difference between two is that if "X" and "Y" are the same logical outcome, then their probabilities are necessarily the same, but an actual person can reply differently depending on how question was formulated.

If you're interested in this subject, I recommend reading about epistemic modal logic - not necessarily for their solutions, but they're clearly aware of this problem, and can describe it better than me.