I think Jaynes more or less defines 'Bayesian methods' to be those gadgets which fall out of the Cox-Polya desiderata (i.e. probability theory as extended logic). Actually, this can't be the whole story given the following quote on page xxiii:

"It is true that all 'Bayesian' calculations are included automatically as particular cases of our rules; but so are all 'frequentist' calculations. Nevertheless, our basic rules are broader than either of these."

In any case, Maximum entropy gives you the pre-Bayesian ensemble (I got that word from here) which then allow the Bayesian crank to turn. In particular, I think Maximum entropy methods are not Bayesian in the sense that they do not follow from the Cox-Polya desiderata.