I would phrase it that astronauts *feel* weightless, because the
sensation of "weight" is really the reaction force of the floor against
our bodies. It is this reaction force that is now missing since both the
person and the "floor" are now in free fall. Thus the person *feels*
weightless, but still has the same weight (suitably reduced due to
increase of R).
I think it is near impossible to make a sensible and consistent
definition of weight other than the gravitational attraction of the
Earth. But one cannot feel a gravitational force (a central and
surprising point to most students). If (uniform) gravitation is the only
force acting, we feel nothing because we are in free fall. We feel only
other forces, we cannot sense a gravitational force which is uniform
over our bodies.