while we still disagree about whether we can have confidence in statements about the outcome of rolling a hundred-sided die.

Ok. I'll attempt to illustrate confidence vs probability as I understand it.

Lets start with your example. Starting with the certain knowledge that there is an object which is a 100-sided die, you are correct to infer that P(roll(D) != 12 | D=100) = 99/100.

Further, you are correct (in this example) to have complete confidence in that estimate.

We can think of confidence as how closely one's probability estimate approaches the true frequency if we iterated the experiment to infinity, or alternatively summed across the multiverse.

If we roll that die an infinite number of times, (or summed across the multiverse), the observed frequency of (roll(D) != 12 | D=100) is more or less guaranteed to converge to the probability estimate of 99%. This is thus a high confidence estimate.

But this high confidence is conditional on your knowledge (and really, your confidence in this knowledge) that there is a die, and the die has 100 sides, and the die is fair, and so on.

Now if you remove all this knowledge, the situation changes dramatically.

Imagine that you know only that there is a die, but not how many sides the die has. You could still make some sort of an estimate. You could guesstimate using your brain's internal heuristics, which wouldn't be so terrible, or you could research dice and make a more informed prior about the unknown number of sides.

From that you might make an informed estimate of 98.7% for P(roll(D) != 12 | D = ???), but this will be a low confidence estimate. In fact once we roll this unknown die a large number of times, we can be fairly certain that the observed frequency will not converge to 98.7%.

So that is the difference between probability and confidence, at least in intuitive english. There are several more concrete algorithmic schemes for dealing with confidence or epistemic uncertainty, but that's the general idea. (We could even take it up a whole new meta level by considering probability distributions of probability functions (one for each possible die type), and this would be a more accurate model, but it is of course no more confident)