I like John's suggestions on what to call g. By the way, Serway's
fourth edition uses "freefall acceleration" instead of "acceleration
due to gravity", which is a big improvement as John notes. I'm quite
impressed that Serway has made an effort to improve his language,
here and elsewhere, even if some of the improvements are just patches.
As for what to call g in the second context, John's "local gravitational
constant" is the best I've heard. I've been saying "force per unit
mass due to gravity", which is a mouthful, even if it has the advantage
of reminding students that the units are N/kg. Of course, there are
a lot of simple dynamics problems where the answer ends up being something
dimensionless times g, and at that last stage, you want to again remember
that g has units of acceleration...