I believe that we agree on the physics but may disagree on the
definition of terms and what is useful pedagogically. I think it is
pedagogically best to use the term "weight" for the gravitational pull
of the Earth. One has to be sure to carefully distinguish this, however,
from the sensation of weight and what reads on a scale (these depend on
a whole host of other forces and accelerations). I do not claim that
this is the only way to teach it. As always, being really careful and
going over enough examples is crucial. These concepts are far more
difficult for students tha we usually realize.