I think I am picking up on three schools of thought (and some variations
on them) in this discussion:
1. The weight of an object is the GMm/r^2 determined gravitational force
on the object. Some folks call this the true weight. The concept of a
force only has meaning in an inertial reference frame. I think that
John Denker is addressing this viewpoint in http://www.av8n.com/physics/weight.htm
when he writes (what I render here in plain text): ** Purists might be
tempted to define a "purely" gravitational weight in terms of [bold,
italics] F_n instead of [bold, italics] F_g ... but this would be
unconventional and very impractical. **
(where F_n is GMm/r^2 and F_g is the mass times the acceleration of the
object when there are no non-gravitational forces acting on it).
2. The weight of an object is the vector sum of a GMm/r^2 contribution
and what some folks call the net pseudo force acting on the object,
where the pseudo force is the force that manifests itself only when the
object is being observed from a non-inertial reference frame. In this
view, the weight of an object can depend on its velocity.
3. There is no such thing as a gravitational force, rather, a more
modern definition of inertia would be an object's inherent tendency to
accelerate in accord with the geometry of space. The weight of the
object is the mass of the object times the acceleration of the object
(as viewed from a specified reference frame which is, in general, not an
inertial reference frame) when the net force on the object is zero.