The reference system for "apparent weight" and "weight" are different. Apparent weight is measured from a non-inertial reference system. Weight is measured from an inertial reference system. Note the definition says "specified" reference system.
italia,
Where in that description does it use the term "Apparent weight" or specify that the frame of reference must be inertial? Once again let me ask- what would you use to measure weight that would distinguish between "Real" and "Apparent" weight?