Your first claim needs qualifications: You should only bet if you're being drawn randomly from everyone. If it is known that one random person in a green room will be asked to bet, then if you wake up in a green room and are asked to bet you should refuse.

P(Heads | you are in a green room) = 0.9 P(Being asked | Heads and Green) = 1/18, P(Being asked | Tails and Green) = 1/2 Hence P(Heads | you are asked in a green room) = 0.5

Of course the OP doesn't choose a random individual to ask, or even a random individual in a green room. The OP asks all people in green rooms in this world.

If there is confusion about when your decision algorithm "chooses", then TDT/UDT can try to make the latter two cases equivalent, by thinking about the "other choices I force". Of course the fact that this asserts some variety of choice for a special individual and not for others, when the situation is symmetric, suggests something is being missed.

What is being missed, to my mind, is a distinction between the distribution of (random individuals | data is observed), and the distribution of (random worlds | data is observed).

In the OP, the latter distribution isn't altered by the update as the observed data occurs somewhere with probability 1 in both cases. The former is because it cares about the number of copies in the two cases.