Who precommits first wins. If the baron precommits to fulfil the threat unless he gets the money, later precommitment of the countess is worthless, since she expects the baron to fulfil the threat anyway.

Obviously. Hence my use of perfect tense rather than present tense. A world with agents acting and reflecting in the way the two players acting in the example do, but without previous commitments that make this precise behavior impossible seems highly implausible to me. I personally would have considered myself as being precommitted not to respond to blackmail in the scenario given even before reading it, and that would have been obvious to anyone familiar enough with me to reasonably feel as confident about predicting my reaction as would be required in the scenario.