Thanks for posting that.
Center of the earth or center of gravity makes no difference. The equipotential surface which is what the figure @GerryBags posted shows is not at a constant distance from either. The surface is determined by the vector sum of all the mass in the universe relative to a particular point on the equipotential surface.
The moon orbits the center of gravity of the earth. But it doesn't change the center of gravity. It *does* change the equipotential surface. They are called tides.
What a spirit level does is show a straight line approximation to the equipotential surface. A plumb bob shows a perpendicular to that surface.
The center of gravity is the point at which mathematically an infinitely dense mass would have the same effect upon other bodies.
In general the distinctions don't matter unless one is involved in geodesy or doing gravity mapping.
I apologize for my poor explanation. I am a reflection seismologist, not a potential fields geophysicist. In industry gravity and magnetics are generally combined because of the mathematical similarities. So while I understand the subject, I'm not practiced at explaining it.
FWIW When I first learned that there was a bump on the surface of the ocean over a seamount it really blew my mind. I know it's true and I understand why it's true, but it is so counterintuitive I still struggle with the idea.