I think you've covered the basics pretty well. The distribution defines all probabilities and moments with regard to relating one with another. correlation only deals with one kind of expectation (i.e. moment) but doesn't imply probabilistic dependence.
In short probability extends to all moments but it's not the reverse (and it's one reason why un-correlatedness does not imply independence but correlation implies dependence). As long as P(A and B) != P(A)P(B) for general events A and B, then there is a dependency between those two events.