> Naturally, large range uncertainty increases the ambiguity of position, but the relative position of the satellites also matters. If they aren’t well spread, the exactness of calculated location also suffers.

(see the excellent example in OP)

Fun tidbit, the resulting error is known for the system in closed form as Geometric Dilution of Precision, and is a 3x3 (edit: or 4+x4+ if you are estimating bias or quantities like time, thx brandmeyer) matrix that depends on all the locations of the visible sats, and your position relative to them.

GDOP is a general relationship for any estimator based only on the equations used to derive something from sensor remote sensor measurements. It's possible to derive GDOP for any sensing system using the Fisher Information Matrix (which is the inverse of GDOP). Some minor caveats apply, but in general this is a useful trick.

FIM is worth learning if you want to get into sensing & estimation. https://en.wikipedia.org/wiki/Fisher_information

Another fun thing: FIM can be derived a number of ways, and appears if you simply ask (mathematically) "What is the most likely position of the gps sensor given sat locations" as the hessian matrix of the system that you use while answering that question using e.g., convex minimization.

All of sensing & estimation is just mostly convex optimization.