What is the difference between no-disturbance principle and non-contextuality? The no-disturbance (ND) principle states that, for any three observables A, B, and C such that A and B are compatible, and A and C are compatible, the probabilities of outcomes of A do not depend on whether A was measured with B or with C.
As far as I know, the definition of non-contextuality is similar to the ND principle. People always take above statement as an example to introduce what is non-contextuality. But we all know quantum physics is contextual. It seems that contextuality contradicts the principle of no-disturbance.
So, what is the difference between them ? Why is the ND principle a more fundamental hypothesis ?
 A: The situation is akin to (in fact, generalizes) the difference between "no-signalling" and "locality" in a Bell experiment.
Generalized no-signalling (sometimes called no-disturbance) means, as the OP suggests, that the probabilities distribution on outcomes of a set of compatible observables is the same regardless of what other observables are measured together with it. In other words, when two contexts overlap (have common observables), the marginals of the observed probability distributions to this overlap are the same.
Non-contextuality means that there is a probability distribution on the outcomes of all the observables at the same time whose marginals yield the observed probability distributions on each context of compatible observables. Another equivalent way to think of non-contextuality is that the observations are consistent with a probability distribution over a hidden variable space where each hidden variable determines pre-existent (deterministic) outcomes for all observables that do not depend on the context that will be measured. 
