Physics for Scientists and Engineers, Tipler, 3rd
Edition page 84 discusses apparent weight and
weightlessness as you have been using these terms.
Defining weight as mg and mMG/r^2 is common practice
in high school and first year college level physics
texts. Engineering Statics and Dynamics texts also
use these definitions for weight. You are not alone
in your views, at least at the level you are teaching.
In my view the free body diagram on the astronaut
shows one arrow pointing
toward the Earth. It is the weight of the astronaut,
or mg (g = 8.7 up
there). The astronaut is in free fall, and they have
plenty of weight. It
is this force which makes them accelerate in orbit.
But the scale reads
zero, so they have no apparent weight.
This is more clear to me, but we each have our own way
of explaining
things.
Anybody out there with a different way of explaining
"weightless"
astronauts?