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I think you've introduced a 'third' definition of weight for which I would
guess there is little support. The choices as I understand them:
1) Weight is the gravitational attraction of the earth on a massive object.
(If the object is on the moon or Mars, substitute the moon or Mars for the
earth.) This definition leads to the discussion of apparent weight such as
an 'apparent weightlessness in orbit.'
2) Weight is the Net force as viewed within the frame of reference that
causes the contact between an object and what supports it.
For someone
standing on a scale, this is still a downward force (since the scale
actually reads the upwards force of the scale) ...
... but includes the CENTRIFUGAL
component due to the earth's rotation, and any effects due to other
gravitational attractors. Because this view looks at the centrifugal force
as 'real' in the frame of reference where the object is at rest, then the
actual weight in orbit is zero.
3) The now introduced definition of weight as the downward force of an
object on it's support--a contact force. I'm not sure what the source of
that force is supposed to be if not (1).