RCCP (Reichenbach's Common Cause Principle) and confounder definition RCCP states that if X and Y are statistically dependent, then there exists Z causally influencing
both.
I've heard a variation of RCCP that states that if X and Y are statistically dependent, one of the following scenarios is possible

*

*Z = X

*Z = Y

*Z is a confounder that is different from X and Y (Z ≠ Y, Z ≠ X), but causes both X and Y

I think that even Bernhard Schölkopf presented RCCP in this manner.
My question is
Given that X and Y are statistically dependent and Z influences both, Z ≠ Y, Z ≠ X, can we say that Z is a confounder? In other words, are the 2 RCCP formulations written above equivalent?
For example:
If A->B->X and A->Y, is A a confounder? (note that A doesn't directly cause X because of mediator B)
In the case above, what can allow us to define the combination of A and B with Z and state that Z is, indeed, the confounder?
 A: A confounder, by definition, is a variable or set of variables that sets up a backdoor path. So if $Z$ causally affects both $X$ and $Y,$ and you are interested in the causal effect of $X$ on $Y,$ then $Z$ is a confounder. The causal diagram would be this:

So the answer to your question

Given that $X$ and $Y$ are statistically dependent and $Z$ influences both, $Z \not= Y, Z \not= X,$ can we say that $Z$ is a confounder?

would be "Yes."
In your example of $A\to B\to X$ and $A\to Y,$ you neglect to mention whether there is an arrow $X\to Y.$ If there is, and I would assume so, then the variable set $Z=\{A,B\}$ is a confounder, because it sets up a backdoor path from $X$ to $Y.$ The principle that lets us define $Z$ in this way is simply the principle of naming things. If we consolidate $\{A,B\}$ down to a single variable $Z,$ then the causal diagram for your scenario (assuming $X\to Y$) would be identical to what's above, because it perfectly captures all the interactions of $Z$ with everything that's not $Z.$
