One of the reasons I like this site is your attention to detail. In that spirit, I would like to suggest that the explanation of how balanced audio works isn't quite accurate. Illustrations really work best to explain this, but I'll try to describe the process as illustratively as I can...
1) In both balanced and unbalanced systems (these can be audio, power, or really anything electrical) the amplitude of the signal is the difference between the voltage of the positive wire (or "hot" or "+") and some reference voltage.
1.u) In unbalanced systems, the reference is the ground, but the ground is always assumed to have zero voltage, even if it doesn't (see 2.u).
1.b) In balanced systems, the reference is the inverted-polarity negative wire (or "cold" or "-"). There is no assumption made as to the voltage of the reference signal because it should be fluctuating equal and opposite to the positive. (Incidentally, each carries .5x the total amplitude of the signal.)
2) In both systems, the ground/shield is intended to intercept as much environmental interference as possible to protect the integrity of the signal on the other wire(s).
2.u) In unbalanced systems, this is the reason it's not a good true zero-voltage reference, and why we have to assume it is zero rather than actually measuring it.
2.b) In balanced signals, the shield/ground is also intended to intercept as much environmental interference as possible. Here it is protecting the integrity of both signal wires, but doesn't serve any reference function.
3) Whatever stray interference enters the signal wire(s) is measured at the receiving end, but how it affects the output signal depends on how the equipment uses the reference.
3.u) In unbalanced systems, the reference is always zero, so any interference that changes the voltage in our one and only signal line becomes part of the signal and is heard/measured as unintended noise.
3.b) In balances systems, because the two signal wires are generally in such close physical proximity, interference will affect both of them in nearly equal measure. Therefore the difference between the "+" and "-" heard/measured at the receiving end will not be significantly altered compared to the source signal.
Let's say we have a 1V amplitude sine wave that we try to distort with the presence of a magnetic field that distorts the signal by .25V (as measured at one instant in time when the source signal is at its peak)...
positive wire = +.5V
negative wire = -.5V
the interference deflects both signals by the same amount...
positive wire = +.5V + .25V = +.75V
negative wire = -.5V + .25V = -.25V
The difference between +.5V and -.5V is the same 1V difference as that between +.75V and -.25V, effectively eliminating the interference. This method is called "common mode rejection" and is the underlying principle of balanced signaling.