If you possess some useful information about the universe you're in, it's magically taken into account by the choice of "information set", because logically, your decision cannot affect the universes that contain copies of you with different states of knowledge,

For example, Counterfactual Mugging: by assumption, your decision logically affects both heads-universe and tails-universe, which (also by assumption) have equal weight, so by agreeing to pay you win more cookies overall. Note that updating on the knowledge that you are in tails-universe (because Omega showed up) doesn't affect anything, because the theory is "updateless".

(I think) I understand the first quote, and (I think) I understand the second quote, but they don't seem to make sense together. If your decision can't affect universes where you have different states of knowledge, why does the decision in the tails-universe affect the heads-universe? Is it because you might be in Omega's simulation in the heads-universe and that therefore there's an instance of you with the same knowledge in both?