I agree with CLiff Parker.
Weight is the force with which the Earth attracts the mass and is the
same in all intertial(!) frames. Frames which are accelerating relative
"to the fixed stars", in Newtonian language, are not inertial frames.
What is measured by a scale, or apparent weight when submerged, is a
different question. NEwton takes for granted that one can define in an
unambiguous way when one is in an inertial frame and one is not.
ANalysis at the end of the last century and into this one showed that it
is not so simple precidely because of the property that gravitational
acceleration is independent of mass, and can be (locally) transformed
away.
From the viewpoint of basic NEwtonian mechanics, weight is just the
force of gravitational attraction, GMm/R^2, in all inertial frames.
Jerry EPstein