I still basically agree with my retracted comment, I'd just like to note that taken at face value, your two equations for B given A really are the same.

The counterfactual difference comes from an implied random variable that decides which branch of the equation we're "using" (in the implied causal process that goes from A to B), and which can remember this information during counterfactual reasoning. But of course it is a simple thing to make this implied random variable an explicit node in your causal graph. This is probably the best resolution.