What’s interesting is that, when you have to decide whether or not to gamble your first coin, the probability is exactly the same in the two cases (p=0.45 of a $2 payout). However, the rational course of action is different. What’s up with that?

That's pretty trivial.

The expected payout of putting a coin into a brown box is 0.90.

The expected payout of putting a coin into a green box is 0.90 plus valuable information about what kind of a green box it is. It is a *different payout*.