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Most elementary physics texts (and students) use the term gravity in two
ways: (A) as a phenomenon which manifests itself (or is responsible for)
weigth or free fall and (B) as a synonyme, sometimes for gravitational
acceleration and sometimes for m*g. I try to emphasize precision but I
am not very pedantic about it.
The principal of equivalence is often stated in textbooks but it is not
emphasized in the first semester. (Many of my students have hard time to
learn the difference between v and a). Same is true or the relative nature
of velocity; we do not always say "with respect to the road". Adding another
ambiguity to the term gravity is not desirable, in my opinion. The
gravitational pull (toward the center of Earth) in the orbiting shuttle, I
say, is about the same as at the sea level but the "apparent weight' is
nearly zero, as in a free-falling elevator.