Re: "If I were the sort of person who rejected the $10, Omega would have told me something else to begin with"
...but why would he do that? Is there some assumption about Omega's motivation here?
It's correct if we expand it to "Omega would have told me something else or not shown up to begin with", or if we're assuming that Omega will show up and say something. It would have to say something like "if you refuse the $10 then the envelope will be empty" — or some other true thing, not the statement given in the original post — since we're assuming it's a perfect predictor and is being honest.
This is a variant built on Gary Drescher's xor problem for timeless decision theory.
You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.
Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.
He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:
"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."
What to do?
EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.