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 “g’s”. It is presumably built into legal issues – for example, you couldn’t 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”.