To make a long story short, I find it helpful to write
the law of universal gravitation in terms of δg, namely:
δg = - rhat ------- 
where rhat is a unit vector in the r direction. The
equation can be used to tell you how much g /changes/
on account of the mass M as you go from the center of
the earth to someplace else, separated by the displacement
If you apply the equation twice, it tells you how
much g /changes/ on account of M as you go from
Spain to New Zealand, which may be more convenient
than trying to make direct measurements at the center
of the earth.
The emphasis here is on the idea of /change/ in g, as
expressed by the δ operator on the LHS of equation .
To say the same thing the other way, it would be a
blunder to leave off the δ operator. Equation 
does not and can not tell you the value of g. The
equivalence principle guarantees that g depends on
the choice of reference frame, whereas the RHS of
equation  is manifestly frame-independent
Let's be clear:
g = acceleration of a free particle, 
relative to the chosen reference frame
Beware that the RHS of equation  is called "gravity"
and the RHS of equation  is also called "gravity". No
wonder students are confused.
Every introductory physics book I've ever seen screws
this up royally ... even the Feynman lectures (which are
in general vastly more reliable than most other books).
This is an example of what I call a bisconception: two
ideas masquerading under the same name.
This sort of thing is exceedingly common. For example,
on a race track, one lap brings you back to the starting
point ... whereas in the pool, you have to swim two laps
to get back to the starting point.
Each definition is correct /in context/. It would be a
mistake to imagine that one is right and the other is
wrong. Instead, *three* ideas are needed: the first
definition, the second definition, and a higher-level
/traffic cop/ idea that indicates which context to use
in any given situation.
An essential first step is to repair the names, perhaps
by adding adjectives.
-- I call equation  the /framative/ gravity. That's
short for "frame-relative". This is denoted g.
-- I call equation  the /massogenic/ gravity. It is
the contribution to g on account of the mass M.
Obviously, the massogenic contribution is not the only
contribution to g.
Less obviously, the massogenic contribution may not even
be the only contribution to δg ... because there are also
centrifugal contributions, if you choose a rotating frame.
The rule is:
Centrifugal fields exist in the rotating frame
and not otherwise. 
In the introductory course, teach everybody rule . It
takes only a moment. Then, if you decide that rotating
frames are outside the scope of the course, that's OK.
To say the same thing the other way, it's not OK to
pretend that such things don't exist. Avoiding them
is a choice, not a law of nature.
You can "mostly" avoid them, but not entirely. The
bisconception exists, and you have to deal with it
at least briefly. Students have seen space-station
videos, and have first-hand experience with cars and
playground equipment where it is perfectly natural
to analyze things in some frame other than the usual
terrestrial lab frame. If you tell them such things
don't exist they simply won't believe you. Even if
you don't want to deal with rotating frames, you still
need the /traffic cop/ to distinguish the two contexts.
And for that matter, the terrestrial lab frame is a
rotating frame. The massogenic contribution is the
main contribution to g, but there are also centrifugal
terms that affect the magnitude and direction of g to
a readily-measurable extent. If you try to calculate
g using equation  alone, you will get the wrong
Everything aboard the space station is really-and-truly
weightless (to an excellent approximation) ... relative
to the frame comoving with the station. The equivalence
principle guarantees this frame is as valid as any other.
To say the same thing the other way, it is a bad idea
to introduce notions of "apparent" weight or "apparent"
weightlessness. The weight is frame-dependent, but it's
not illusory or imaginary or merely apparent.