I think about planes as being defined by a normal n and a point on the plane p. Then we can define the plane as (read * as dot-product):
P: n * x - n * p = n * ( x - p ) = 0
We can now define d = n * p and get:
P: n * x - d = 0
I thinks this is the more formal definition that you would find it in a textbook. If you use 4D vectors for planes and want to define the distance of a point to a plane using the dot product you would define d = -n * p which yields:
P : n * x + d = 0
I think the 4D vector definition also works well with transformations where you simply multiply the 4D 'plane' vector with a 4x4 transformation matrix (not sure though). Personally I prefer with the more formal definition and use explicit plane functions to evaluate the distance to planes or transform them. If you want to wrap everything into a generic 4D vector the later might be the better choice.