Whenever an aircraft is certified, the manufacturer has to proof the kind and extension of the wake turbulences; and the result of these test also are part of the given category.
Another example can be the Boeing 757. If preceeding other traffic, this aircraft is considered to be "heavy", when following another aircraft, the 757 is "only" medium and the ATC controller has to maintain a lower separation.
In your list, "J" for "super" is missing, a category, actually only given to the A380.