That makes the two cases, stated as they are, different. In game 1 the utility of U1($0) has negative value: a sense of guilt (or shame) over having made the bad choice, which doesn’t seem possible in game 2 (because game 2 is stated in terms of abstract probabilities, see below).
If you could choose whether or not to have this guilt, would you choose to have it? Does it make you better off?
If you could choose whether or not to have this guilt, would you choose to have it? Does it make you better off?