Is it that the coin flip is a random process, but that the ball may have gotten into the urn by some deterministic method?

Randomness is uncertainty, and determinism doesn't absolve you of uncertainty. If you find yourself wondering what exactly was that deterministic process that fits your incomplete knowledge, it is a thought about randomness. A coin flip is as random as a pre-placed ball in an urn, both in deterministic and stochastic worlds, so long as you don't know what the outcome is, based on the given state of knowledge.

Is it that we have partial information about the urn state, therefore the odds will not be 50-50, but potentially something else?

The tricky part is what this "partial information" is, as, for example, looking at the urn after Omega reveals the actual color of the ball doesn't count.

Another difference is that in the original problem, the positive payoff was much larger than the negative one, while in this case, they are equal. Is that significant?

In the original problem, payoffs differ so much to counteract lack of identity between amount of money and utility, so that the bet does look better than nothing. For example, even if $100*0.5-$100*0.5>0, it doesn't guarantee that U($100)*0.5+U(-$100)*0.5>0. In this post, the values in utilons are substituted directly to place the 50/50 bet exactly at neutral.

And once again, if this were not an Omega question, but just some random person offering a deal whose outcome depended on a coin flip vs a coin in an urn, why don't the same considerations arise?

They could, you'd just need to compute that tricky answer that is the topic of this post, to close the deal. This question actually appears in legal practice, see hindsight bias.