I think you are right about the sportsball case! I've updated my meta-meta-probability curve accordingly :-)

The wikipedia article on the Beta distribution has a good discussion of possible priors to use. The Jeffreys prior is probably the one I'd use for Sportsball, but the Bayes-Laplace prior is generally acceptable as a representation of ignorance.

The example I like to give is the uncertain digital coin- I generate some double p between 0 and 1 using a random number generator, and then write a function "flip" which generates another double, and compares it to p. This is analogous to your blue box, and if you're confident in the RNG means you have a tight meta-meta-probability curve, which justifies the uniform prior.

Jaynes uses "the probability that there was once life on Mars" in his discussion of this. I'm not sure that's such a great example either.

Yeah, that seems like a good candidate for the Haldane prior to me.