The error in the reasoning is that it is not you who makes the decision, but the COD (collective of the deciders), which might be composed of different individuals in each round and might be one or nine depending on the coin toss.

In every round the COD will get told that they are deciders but they don't get any new information because this was already known beforehand.

P(T| you are told that you are a decider) = 0.9

P(T| COD is told that COD is the decider) = P(T) = 0.5

To make it easier to understand why the "yes" strategy is wrong, if you say yes every time, you will only be wrong on average once every 9 turns, the one time where the coin comes up head and you are the sole decider. This sounds like a good strategy until you realize that every time the coin comes up head another one(on average) will be the sole decider and make the wrong choice by saying yes. So the COD will end up with 0.5*1000 + 0.5*100 = 550 expected donation.