How is it different? Because in one case there is a "rightful" owner, and the first case, the money appears to be "shared" at one point in time?
Very precisely. You think that shouldn't matter?
That is also why, in a subtle sense, db's framing makes the problem different, even though the underlying reward matrix remains identical in monetary terms.
I have insisted you
don't necessarily treat this as a game theoretical abstract problem where utility is perfectly encoded in the monetary payoff and your job is just to maximize that. That is, not unless that's how you honestly would frame it in real life.
The purpose of the thought experiment is exactly to explore how real human emotions and perceptions, and how real social dynamics, transform a problem which otherwise (that is, in an abstract game theoretical formulation) would have a clear unambiguous solution, invariant from all the different framings that maintain the same monetary reward structure. That is, to explore the difference between rationality as formally defined in game theory and what makes sense to you in real life.
Whether you choose to call the latter "rational" or not is a matter of definitional nuance and not the main point.
Any definition of rationality that only takes into account monetary incentives is completely flawed. Money is a means to some further ends, something that will satisfy some other human desire. If you're poor and starving, I'd imagine you would take any amount because you value your next meal over any chance to exact vengeance. On the other hand, if you have everything in the world, what's $10 over the chance to share your views on cooperation/vengeance with someone that just seemingly slighted you? The psychological satisfaction that results would be worth more than another $10 in your pocket.