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Quoting "Rauber, Joel" <Joel.Rauber@SDSTATE.EDU>:
It is a valid definition IMO, not my preference to define the weight
as m*g where g is
M*g = GMm/r^2 direction presumably pointing towards the center of a
uniform spherical planet.
Strictly, this is still not true.
The direction of g is not towards the centre of a uniform spherical planet.
Note that there is also gravitational force on the object from the
moon, Sun, etc...
Some may even insist on including dark matter and dark energy. :-)