Hi Mauro,
I am flattered that you think I have philosophical training. I don't. I just read a lot of books about the philosophy of probability.
If you are a follower of Cox then it is definitely true that there is a wide gap between your position and mine. I am a subjective Bayesian in the vein of Ramsey, de Finetti, Savage, Jeffrey et. al. and I think that Cox's derivation of probability theory is one of the silliest things I have ever seen. Debating the relative merits of the two approaches could occupy a lot of space, so I will confine myself to a couple of comments.
Firstly, Cox's approach contains a lot of arbitrariness. For example, he starts from the idea that degrees of belief have to be represented by real numbers, with no real justification other than simplicity. Why do they have to be totally ordered rather than just partially ordered? Weak analogies with measuring distance with a ruler just don't cut it for me, especially since the approach does not explain how one would construct an analogous device for measuring someone's belief that would yield a real number.
Secondly, and relatedly, I believe that a viable approach to the foundations of probability has to be operational, i.e. it must say what things in the world correspond to probabilities and how to measure them. Subjective Bayesianism does this, i.e. it explains how to measure probabilities in terms of an agent's actions, but no other approach to probability really does. It is a bit complicated to explain why I think operationalism is needed here given that I am not an operationalist. Indeed, I don't actually think that probabilities ultimately should be defined in a purely operational way. It is just that, when you are confused about why a theory works, i.e. you cannot quite derive the results you need to justify the way it is applied, then it is a good idea to try to analyse the problematic concept in terms of something else and then use agreed upon facts about that other thing to see if you can find a better justification. Directly measurable things are the type of things about which we have a lot of agreed upon facts that anyone can verify, so operational definitions are the most useful for this purpose. I don't view operational definitions as "the" definition of the concept in question, but they provide a very useful rigging when there is a controversy to be resolved. As an aside, this is how I reconcile Einstein's approach to special relativity with his later statements on physics. It is not that he wanted to define spacetime operationally, but rather that he knew something had to change about the nature of space and time. The concepts of space and time come in a tight package with all the rest of the concepts of classical physics and it is very difficult to see how to unpick that package when you want to make some fundamental change. One way of getting around this is to redefine the problematic concepts, temporarily, in an operational fashion. However, after we are finished we can go back to being straightforwardly realist, e.g. viewing the structure of spacetime as the fundamental thing that accounts for the way that light rays behave rather than the other way around. It is the same with probability. We can't agree why statistics works so there must be something wrong with our usual concepts and derivations. However, probabilities are tied up with the whole theory in a tight package so it is best to temporarily define them in terms of something directly measurable. By the way, in the context of quantum theory, I think this is what Lucien means when he says that we should adopt an "operational methodology" without necessarily being operationalists.
Regarding the meaning of "context", I presume you understand that in quantum theory I intend it to be synonymous with the choice of measurement. In general, a context is the thing that determines the set of bets that can be jointly resolved. Now, of course, if we already have probability theory then we could say that there is a probability for each context and then a conditional probability for each measurement outcome given the context. Multiply the two together and you have a joint probability distribution over contexts and outcomes, which is just an ordinary classical distribution. However, the point is that we are trying to derive probability theory rather than assuming it so we have to ask what would force our beliefs about the context to be described by a classical probability distribution. I suppose you could write down an exhaustive list of all contexts and then allow bets to be made on the context as well as the measurement outcomes. Then you could apply a Dutch book to the bets on context. That would be reasonable in the 20 questions game the way I have described it in which a third party is doing the questioning. However, I also want to allow for the possibility that the bookie might be the person choosing the context and they might choose the context adversarially after you have announced your probabilities (or similarly it might be you choosing the context after making your bets and putting the bookie at a disadvantage). It might have been clearer if I had described things this way in the essay. In this case, the choice of context is not something that you can assign a probability to. Instead, you have to do a worst case analysis and hedge against all possible contexts. This type of setup is the Bayesian way of fleshing out what it means for the choice of context to be a "free choice" that we cannot assign probabilities to. Practically it just means that it might be determined adversarially so we have to do a worst case analysis.
Regarding many-worlds, I do not currently think it is a "realistic approach", but hopefully we can agree that it is a realist one (important distinction there). Although I do not advocate the theory, it remains the only interpretation of quantum theory in which a fully subjective Bayesian derivation of the Born rule along the lines I suggest has been carried out, so it would be unfair of me not to mention it. However, it is not too surprising that they are able to do this, since they start from the premise that the quantum state is real and that is the thing that carries all the information about the probabilities in the first place. It would not be too hard to derive classical probability theory if you started from the premise that reality was described by an object isomorphic to a probability distribution, and I hope we would all reject such a derivation as silly. As it happens, I am toying with a version of many worlds in which the wavefunction is not real but I still think you can derive the Born rule. I am not taking this too seriously, since it is just meant as a counterexample to the PBR theorem showing that you can have a realist theory with an epistemic quantum state if you broaden the ontology in some way. I don't think many-worlds is the best way of broadening the ontology, but one has to start somewhere and it is a more concrete suggestion than vague talk about retrocausality or "relational degrees of freedom" that you might hear from me and Rob Spekkens on other days.