This is called Granger-causality (and work on it led to a Nobel prize, so it's important and useful)... it's stronger than just correlation, and way easier to determine than true causation, but it's possible that z causes both x and y, and z's effect on x is just more delayed than its effect on y.
But it at least rules out x causing y, which is something.
> but it's possible that z causes both x and y, and z's effect on x is just more delayed than its effect on y.
This is in fact the case with the barometer falling before a storm. Both the falling barometer and the subsequent rain and wind of a storm are consequences of an uneven distribution of heat and moisture in the atmosphere approaching equilibrium under the constraints of Earth's gravity and Coriolis force.
Still doesn't work. Suppose I flip a coin and write the result in two places. I write it on sheet y then sheet x. We have that X == Y, so p(x|y) = 1, p(x|!y) = 0, and time(y) < time(x), but neither causes the other. I can write more later if you have interest, but I gotta run.
p(x|y) = 1, p(x|!y) = 0, and time(y) < time(x)
That rules out the rooster counter-example. If y is a boolean, I guess the only thing you can "do" to it is negate it.