That argument just shows that, in the second betting scenario, Beauty should say that her probability of Heads is 1⁄2. It doesn’t show that Beauty’s actual internal probability of Heads should be 1⁄2. She’s incentivized to lie.
EDIT: Actually, on considering further, Beauty probably should not say that her probability of Heads is 1⁄2. She should probably use a randomized strategy, picking what she says from some distribution (independently for each wakening). The distribution to use would depend on the details of what the bet/bets is/are.
That argument just shows that, in the second betting scenario, Beauty should say that her probability of Heads is 1⁄2. It doesn’t show that Beauty’s actual internal probability of Heads should be 1⁄2. She’s incentivized to lie.
EDIT: Actually, on considering further, Beauty probably should not say that her probability of Heads is 1⁄2. She should probably use a randomized strategy, picking what she says from some distribution (independently for each wakening). The distribution to use would depend on the details of what the bet/bets is/are.