What tests can I perform to figure out if a wire is my Grounding Electrode Conductor?
I was taught to step back and take a good look at what I was trying to accomplish. The two reasons for grounding are:


To limit voltages caused by lighting or by accidental contact of the supply conductors with conductors of higher voltage.
To stabilize the the voltage under normal operating conditions (which maintains the voltage at one level relative to ground, so that any equipment connected to the system will be subject only to that potential difference).


That's right out of the Handbook. 

Personally I can't think of a better grounding electrode that a metal well pipe sunk over 30' in the ground, and even though some of the system is possibly not exactly code (the #10 ground) any ground is better than no ground. I think if I could verify that the grounding conductors that you mentioned were actually part of the grounding electrode conductors, then I would make any necessary corrections and use it. You are allowed to tie grounding electrode conductors together other than the panel so long as they maintain a single point of ground. 

One final comment, reading and studying NEC Article 210 all these years, I believe that saying there is redundancy in the grounding system is from the department of redundancy department (joke).

Hope this helps and good luck.