As long as we’re modeling things in threespace, mapped onto a sphere, we might as well throw elevation data in there. Even if we still assume a uniform thickness, there’s more material of that uniform thickness needed in order to construct a square mile of mountainous terrain versus a square mile of prairie.
And of course we’d get yet another answer if we didn’t assume uniform thickness, but rather weighed in all the material down to the center of the earth. Take the solid formed by cutting at the borders all the way to the center of the earth (down to a point) and find the center of gravity of that object. The spot directly above this CG is the answer.
Wait, do we assume uniform density too? There’s gold in some of them thar hills, you know.
There are dozens of different ways you could reasonably define “center” that would come up with different answers. How about, “the spot that is the furthest from any border” for a potentially wildly different one?
Well, you also get different results from caveman’s method #5 and the method where you’d draw a vertical line halfway between the eastern and western boundaries, and a horizontal line halfway between the northern and southern boundaries and find their intersection.
But I favor methods which would give the same results no matter which way you oriented the coordinate system, which neither of those methods would do.
BTW, it’s easy to come up with a trapezoid where, using caveman’s method #5, the “center” is at one of the corners.
In the book Tuva or Bust! (highlighting the late Richard Feynmann’s quest to visit that storied land in Central Asia), he and Ralph Leighton calculated the “center of Asia” by the balancing method. There is a monument in Tuva marking the spot, though it is not the same place Feymann and Leighton calculated.