In what way, if any, is this problem importantly different from the following "less mathy" problem?

You have a sealed box containing a loose coin. You shake the box and then set it on the table. There is no a priori reason for you to think that the coin is more or less likely to have landed heads than tails. You then take a test, which includes the question: "Did the coin land heads?" Fortunately, you have a scanning device, which you can point at the box and which will tell you whether the coin landed heads or tails. Unfortunately, the opaque box presents some difficulty even to the scanning device, so the device's answer is right only 99% of the time. Furthermore, its errors are stochastic (or even involve quantum randomness), so, for any given coin-in-a-box, the device is probably correct but has a chance of making an error. You point the scanning device at the box and observe the result (it's "heads").

Then, unsurprisingly, Omega appears and presents you with the following decision. Consider the counterfactual world where the coin landed the same as it did in your world, but where the scanning device displayed "tails" instead of "heads", after you pointed it at the box. You are to determine what Omega writes on the test sheet in that counterfactual world.