if you know the states of all the molecules in a glass of hot water, it is cold in a genuinely thermodynamic sense: you can take electricity out of it and leave behind an ice cube.

I am not sure this is true as stated. An omniscient Maxwell demon that would only allow hot molecules out runs into a number of problems, and an experimentally constructed Maxwell's demon works by converting coherent light (low entropy) into incoherent (high entropy).