It is equivalent: you can also calculate the two partial derivatives, take the cross product and then the norm:
twoPartialDerivatives = {D[r, x], D[r, y]}
crossProduct = Cross @@ twoPartialDerivatives
areaElement =
Simplify[Norm[crossProduct],
Element[x | y, Reals]]
The version with the jacobian is more easily genralized to higher dimensions. You will find textbooks that use this approach.
