As usual, I have my...issues...with dice as Props, but I will leave those aside for the good of the thread.
I do think there's space for one more area wherein dice matter, which doesn't fit well into any of the existing categories: Dice (or other RNG sources) as Gradients. That is, dice give us the possibility of having more than a binary outcome, more than just yes/no, win/lose, etc. Damage dice are obviously one of the most common examples of gradient dice, but PbtA games have their three grades of success (which I personally have extended to a fourth grade), and Pathfinder 2e has a more clear, direct form of fumble/fail/pass/crit than is typical in d20 systems. Those are both examples that combine some form of prompt (success vs failure) with gradient (degrees of success.) In most cases like this, the tone and focus of the action is already defined, but exactly how it cashes out remains in question, unlike a usual "prompt"-style role, where there are numerous completely disparate possible results (e.g. rolling up an NPC with various quirks or rolling up a random treasure.)
Gradient stuff doesn't see as much presence in D20 games, but is rather more significant in, say, White Wolf's Storyteller games or in the various editions of Shadowrun, where you have dice pools, and usually need to hit a certain number of successes (and may get more benefits for more successes.)