There has been recent interest in extending the concept of contextuality to cases of disturbance or inconsistent connectedness. This talk will describe an approach using probabilistic causal models, which generalize the hidden-variables models of Bell and Kochen & Specker, following recent work by Cavalcanti. I first prove an equivalence between three conditions on an arbitrary measurement system: (1) existence of a model minimizing all causal influences of context upon measurement outcomes, (2) prohibition of a form of "hidden" causal influence, and (3) noncontextuality as defined in the Contextuality-by-Default (CbD) theory of Dzhafarov and Kujala. The no-hidden-influence principle thus confers a physical interpretation to CbD-contextuality, paralleling Bell's local causality and Kochen & Specker's classical embeddability. I then extend this analysis to other causal graph topologies, showing that different graphs yield different notions of contextuality, but only the one corresponding to CbD agrees with traditional contextuality when restricted to non-disturbing systems.