Sorry for not being clear. The d-separation criterion is the same in both Bayesian networks and causal diagrams, but its meaning is not the same. This is because an arrow A -> B in a causal diagram means (loosely) that A is a direct cause of B at the level of granularity of the model, while an arrow A -> B in a Bayesian network has a more complicated to explain meaning having to do with the Markov factorization and conditional independence. D-separation talks about arrows in both cases, but asserts different things due to a difference in the meaning of those arrows.

A Bayesian network model is just a statistical model (a set of joint distributions) associated with a directed acyclic graph. Specifically it's all distributions p(x1, ..., xk) that factorize as a product of terms of the form p(xi | parents(xi)). Nothing more, nothing less. Nothing about causality in that definition.

I think examples for (1),(2),(3) are simpler than Manfred's Berkson's bias example.

(1) A <- C -> B

Most clearly non-causal associations go here: "shoe size correlates with IQ" and its kin.

(2) A -> C -> B, and (3) A <- C <- B

Classic scientific triumphs go here: "smoking causes cancer." Of note here is that if we can find an observable unconfounded C that intercepts all/most of the causal pathway, this is extremely valuable for estimating effects. If you can design an experiment with such a C, you don't even have to randomize A.