I disagree with the part "her actions lead to different outcomes depending on what day it is." The way I see it, the "outcome" is the state of the entire multiverse. It doesn't depend on "what day it is" since "it" is undefined. The sleeping beauty's action simultaneously affects the multiverse through several "points of interaction" which are located in different days.
What makes something an "outcome" in Savage's theorem is simply that it follows a certain set of rules and relationships - the interpretation into the real world is left to the reader.
It's totally possible to regard the state of the entire universe as the "outcome" - in that case, the things that corresponds to the "actions" (the thing that the agent chooses between to get different "outcomes") are actually the strategies that the agent could follow. And the thing that the agent always acts as if it has probabilities...
Vladimir Slepnev (aka cousin_it) gives a popular introduction to logical counterfactuals and modal updateless decision theory in the Tel Aviv LessWrong meetup.
[https://www.youtube.com/watch?v=Ad30JlVh4dM&feature=youtu.be]