When I studied physics in highschool, we were taught to measure mass in kilograms and to measure weight in newtons. While this seemed a little strange at the time, it had the effect of forcing you to be clear which one you were talking about and it avoided this kind of confusion.
As I recall, the mass term "slug" is often used rather than pound-mass to remove this ambiguity. A slug = 32.174049 lbm (i.e. multiplied by the earth's gravitational acceleration constant without any units).
But, yes, the terminology among other things can introduce ambiguity and the potential for error when using Imperial units for these types of calculations. (In fact, I seem to remember it sometimes was easier to convert into metric and back again at the end.)
The general point here though is that the local gravitational constant changes the weight (downward force on the object) but not its inertia (a property of mass and other physical properties).
