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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.