There are multiple concepts mentioned here rather than two: geometric centroid, center of mass, and center of gravity, and another "center of gravity" concept where a gravitational field apparently defines it.
Geometric centroid is the furthest off the other two. It is only the same as center of mass if either one is assuming uniform density or some balanced masses just happen to make it so.
Center of mass is a rather interesting concept that is most commonly used. Gravity and weight(balancing) happen to be an easy way to measure this but only because the gravitational field is very uniform relative to the object, but one could also leverage other phenomenon such as forces and moments experienced when rotating an object about various points.
Center of gravity is intended to be the point where the forces due to gravity from the object itself vanishes (aka cancel each other out). Isolating only the forces of gravity of an object due to itself requires bringing in some theoretical models to exclude effects of gravity from other objects. For all that we know, this happens to be the center of mass thanks to the definition of gravitational force. Maybe some day someone will theorize something where the mass and gravity relation needs to be tweaked to maintain all other non-gravity related mass physics, but until then, this center of gravity remains the synonym of the center of mass.
Finally a center of gravitational? force that the poster is calling a center of gravity. Note that by considering gravitational forces from OTHER objects, there is no guarantee that there shall exist a vanishing point of zero gravitational force applied to OUR object. Such is the case for an object of nonzero mass at rest on the ground, considering only all gravity(primarily from the earth) and ignoring the electromagnetic forces that keep it from accelerating down through the ground. The closest definition I can think of would be: points where a force (and zero moment) can be applied to cancel all gravitational forces on the object if the object is assumed to be rigid. With such a specific definition, the only answer I can think of is that it matters when it matters, and it is affected by the scale rather than the planet it is on.
If I'm considering a small object on earth in earth's reference frame, I might be able to draw a line parallel to the gravitational fields, through the center of mass to identify the points. However if I consider just a proton, I may have to compensate for gravity from other protons or neutrons as proximity overtakes mass, or maybe I just don't care about gravity from earth. Maybe my object is such that the primary effects of gravity are due to the sun despite being on earth - object as a part of earth's core (or some other) balanced subsection such that gravity due to earth cancels out.
:. For things that matter consider all forces; discrimination without a good purpose is bad.