Is there any practical difference between "assuming independent results" and "assuming zero probability for all models which do not generate independent results"?

No.

If not then I think we've just been exposed to people using different terminology.

I think it's more than terminology. And if Mencius can be dismissed as someone who does not really get Bayesian inference, one can surely not say the same of Cosma Shalizi, who has made the same argument somewhere on his blog. (It was a few years ago and I can't easily find a link. It might have been in a technical report or a published paper instead.) Suppose a Bayesian is trying to estimate the mean of a normal distribution from incoming data. He has a prior distribution of the mean, and each new observation updates that prior. But what if the data are not drawn from a normal distribution, but from the sum of two such distributions with well separated peaks? The Bayesian (he says) can never discover that. Instead, his estimate of the position of the single peak that he is committed to will wander up and down between the two real peaks, like the Flying Dutchman cursed never to find a port, while the posterior probability of seeing the data that he has seen plummets (on the log-odds scale) towards minus infinity. But he cannot avoid this: no evidence can let him update towards anything his prior gives zero probability to.

What (he says) can save the Bayesian from this fate? Model-checking. Look at the data and see if they are actually consistent with any model in the class you are trying to fit. If not, think of a better model and fit that.

Andrew Gelman says the same; there's a chapter of his book devoted to model checking. And here's a paper by both of them on Bayesian inference and philosophy of science, in which they explicitly describe model-checking as "non-Bayesian checking of Bayesian models". My impression (not being a statistician) is that their view is currently the standard one.

I believe the hard-line Bayesian response to that would be that model checking should itself be a Bayesian process. (I'm distancing myself from this claim, because as a non-statistician, I don't need to have any position on this. I just want to see the position stated here.) The single-peaked prior in Shalizi's story was merely a conditional one: supposing the true distribution to be in that family, the Bayesian estimate does indeed behave in that way. But all we have to do to save the Bayesian from a fate worse than frequentism is to widen the picture. That prior was merely a subset, worked with for computational convenience, but in the true prior, that prior only accounted for some fraction p<1 of the probability mass, the remaining 1-p being assigned to "something else". Then when the data fail to conform to any single Gaussian, the "something else" alternative will eventually overshadow the Gaussian model, and will need to be expanded into more detail.

"But," the soft Bayesians might say, "how do you expand that 'something else' into new models by Bayesian means? You would need a universal prior, a prior whose support includes every possible hypothesis. Where do you get one of those? Solomonoff? Ha! And if what you actually do when your model doesn't fit looks the same as what we do, why pretend it's Bayesian inference?"

I suppose this would be Eliezer's answer to that last question.

I am not persuaded that the harder Bayesians have any more concrete answer. Solmonoff induction is uncomputable and seems to unnaturally favour short hypotheses involving Busy-Beaver-sized numbers. And any computable approximation to it looks to me like brute-forcing an NP-hard problem.