there's no term for "how surprised I was" in Bayes' Theorem.

Not quite. The intuitive notion of "how surprised you were" maps closely to bayesian likelihood ratios.

Regarding your die/beads scenarios:

In your die scenario, you have one highly favored model that assigns equal probability to each possible number. In the beads scenario you have many possible models, all with low probability; averaging their predictions gives equal probability to each possible color.

To simplify things, let's say our only models are M, which predicts the outcomes are random and equally likely (i.e. a fair die or jar filled with an even ratio of 12 colors of beads), and not-M (i.e. a weighted die or jar filled with all the same color beads). In the beads scenario we might guess that P(M)=.1; in the die scenario P(M)=.99. In both cases, our probability of red/one is 1/12, because neither of our models tell us which color/number to expect. But our probability of winning the bet is different -- we only win if M is correct.