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To me, the strange thing about the NIST definition is the terminology
"local force of gravity". Apparently "force of gravity" is NOT well-established!
What's strange about that? The physics requires it. Force is
a vector. The vector direction is clearly different "here" versus
the antipodes. The vector magnitude is a function of altitude.
From NIST:
"The local force of gravity on a body, that is, its weight, consists of the
resultant of all the gravitational forces acting on the body and the local
centrifugal force due to the rotation of the celestial object."
That makes sense to me. That's the definition I've been using
for a long time.
It also agrees with the convention of measuring altitude relative to
sea level. The sea is well approximated as an isopotential if and
only if you include the centrifugal field as well as the GmM/r^2
gravitational interaction.