For about the last 20 years of my university teaching career I NEVER used the term "weight" or the symbol "W" for mg. In free-body force diagrams I'd include the gravitational force mg (as a vector) on the object and call it the gravitational force; g was not necessarily 9.81 N/kg. At most I'd point out that the common use of weight refers to the (electromagnetic) upward force on an object on a surface, and that fits in with the concept of "weightlessness" in an orbiting spaceship: an object on the spaceship has a gravitational force mg acting on it (not the g at the surface of the earth, however), but needs no normal or other force. Also, then, the weight of a person in an accelerating elevator, say, could be different in magnitude from mg. Nothing more on weight. I stopped using regular textbooks in part for this reason, and made my own notes and questions available as pdf files.