I have read quite a lot about how the GPS system works and I am confused by some of the statements made. The GPS signals are time signals and the position is determined from the time difference between the signals when they arrive at the GPS unit.
The distance (time delay) from a satellite puts the GPS somewhere on the surface of a notional sphere.
The flaw in that description is that the receiver doesn't have a good clock, so it has to get its time reference from the satellites. The receivers actually measure the time difference between the signals from the various satellites in view, and calculate the position from the satellites' positions and these time differences.
It may be that the calculation is actually done by guessing time and position from one signal, and then modifying that guess to align with signals from the other satellites. The result is the same, that the time delay is not what matters, it is the difference in time delays that gives the position.
The chess-board analogy shows this. Moving the receiver up and down means the signals arrive sooner or later, but that can't be detected if the satellites are largely overhead. Position can be detected because horizontal movement puts the receiver closer to some satellites and further from others. To get altitude accurately, there need to be satellites that are low, nearer the horizon, visible to the receiver.
Also, on GPS receivers that have external aerials, the position calculated is that of the aerial, not the receiver, and the length of aerial cable makes no difference, unless you want a really good time reference and you are worried about being 0.000000005 seconds out for every metre of aerial cable.