If you have a 0K icecube one one hand and a gas and piece of RAM encoding that state of the gas on the other, then they certainly do not seem to be in equilibrium in this sense:

gas+RAM is not in thermal equilibrium with the ice cube, because a large enough stick of RAM to hold this information would itself have entropy, and a lot of it (far, far larger than the information it is storing). This is actually the reason why Maxwell demons are impossible in practice - storing the information becomes a very difficult problem, and the entropy of the system becomes entirely contained within the storage medium. If the storage medium is assumed to be immaterial (an implicit assumption which we are making in this example), then the total system entropy is 0 and it's at 0 K.

My more general point is that one can not just claim by fiat that the Jaynes-style definition is the "true" one;

It is true for the same reason that Bayesian updating is the only true method for updating beliefs; any other method is either suboptimal or inconsistent or both. In fact it is the very same reason, because the entropy of a physical system is literally the entropy of the Bayesian posterior distribution of the parameters of the system according to some model.