A plane defined as 4 numbers (x,y,z,distance) is known as the hessian normal form, Where the xyz values are unit-length.
However I've found its not necessary to have the xyz unit length (in software I can still perform useful operations on the plane when the xyz vector isn't normalized, though it must be non-zero length).
This may seem obvious since its a standard representation as far as I can tell, but I want to refer to it specifically.
Whats the correct name/term for a plane defined by (x,y,z,distance) which isn't necessarily having a unit-length xyz ?
The thing is, that the plane itself doesn't have a name at all, but the way to define the plane has. The Hessian Normal Form has the form $x\cdot n+d=0$, where $d$ is the distance and $n$ is a normalized normal vector. However, the same plane can also be defined using the Normal Form I gave you in my first comment.
So to answer your question, if the normal vector $n=(x,y,z)$ doesn't have unit length, then you have a plane in "Normal Form", otherwise it's "Hessian Normal Form".