I think that there are (at least) four usages of g floating around this discussion!!!
1) g = a_g = acceleration due to gravity alone
g is simply GM/r^2 with no adjustment for non-inertial frames.
2) g = a_g a_c(earth)
g is GM/r^2 with a correction for the centripetal acceleration due to the rotation of the earth (or by extension, any other planet); it is the acceleration on the surface of earth. When you say g = 9.81 m/s^2, this value starts with a_g and subtracts a little bit for a_c. This value changes slightly in magnitude from point to point on the earth.
3) g = a_f = free fall acceleration
g is adjusted for any non-inertial frames. In an elevator accelerating upward, g is greater than 9.8 m/s^2; in free fall or in orbit, g = 0. Note that this values depends on your reference frame. To the person in the elevator, g > 9.8 m/s^2, but to the person on the ground, g = 9.8 m/s^2
4) g = 9.80665 m/s^2 = the standard acceleration of gravity. NIST uses g_N for this quantity. This is a fixed constant used for conversions to gs. It is presumably built into legal issues for example, you couldnt buy 1 pound of gold on a mountain top, shave off a little, and then sell 1 pound at sea level.
Perhaps we need to ban g in this discussion and instead use the more specific notion. I see posts to the effect when I say weight clearly what I mean is mg and I cringe which g? I have a feeling that half of the debate would disappear if it were clear just which g people mean.
Personally, I would associate Definition 1 of g with force of gravity and Definition 3 with weight.