Understanding conditional independence of two random variables given a third one

The first definition is the informal one, but at the same time seems rather convoluted to me.

I'd prefer: X and Y are conditionally independent with respect to a given Z iff

$P(X \; Y | Z) = P(X | Z ) P(Y | Z)$

Recall that conditioning one (or several) variables on the value of another, is (informally) the same as restricting the whole universe to a part of it. Then, if you are given the value of $Z$, you can think as if you are defining new variables that are the same as the unconditioned but that are restricted to our new (smaller universe) $X' \equiv X | Z$ $Y' \equiv Y | Z$ The above formula simply states that $X'$ and $Y'$ are independent.

The first definition says the same, but applying (in words) the property that two variables are independent iff their conditioned probabilities are the same as the unconditioned : $A$ indep $B$ iff $P(A | B ) = P (A)$