The similarity between "how much noise each string is making" and "how each string interacts with the magnetic field" is greater than your question implies.
In an acoustic guitar
- The vibration of the string transfers through the bridge of the guitar to the sounding board. This saps energy from the string causing its vibration to decay.
- The sounding board transfers its vibration to the air, by virtue of being a big flat surface that pushes and pulls air as it vibrates. This saps energy from the sound board causing its vibration to decay. The sound board has resonances of its own that affect the tone of the sound.
- Other components resonate or absorb vibrations at certain frequencies - the neck via the nut, etc. This subtly affects the overall tone and sustain of the instrument.
In an electric guitar
- The vibration of the string transfers through magnetic induction to the pickup. This saps energy from the string causing its vibration to decay.
- The resulting electric current is carried to effects, amp, speaker. These components have properties of their own that affect the sound.
- Other components resonate or absorb vibrations at certain frequencies - the body of the guitar via the bridge, the neck via the nut, etc. This subtly affects the overall tone and sustain of the instrument.
The actual vibration of an acoustic string versus an electric string isn't much different. A mathematically "ideal" electric guitar would be completely rigid, so that no energy is lost other than through the pickup - and solid-body guitars come closest to that. Because the loudness of the amplified sound comes from another source of energy, a pickup takes very little energy from the string, so electric guitars can have long sustain.
Other than that, the string behaves the same on both acoustics and electrics. Thick heavy strings vibrate slowly for a bass sound, light thin strings vibrate quickly for a treble sound. If you pluck near the middle of the string you get close to a pure sine wave. If you pluck away from the centre you get more layers of vibration: harmonics.
You use the word "unbalanced", but I'm not sure what you mean by it.
Typically when talking about electrical sound signals, "unbalanced" refers to a signal path that doesn't compensate for interference. If I ran 100 metres of guitar cable -- which contains a signal wire and a ground wire -- and plugged it into a guitar and an amp, I would hear a load of crackle and hum, just from stray magnetic interference in the environment.
If I ran 100 metres of balanced microphone cable in the same way, to suitable equipment, I wouldn't hear this interference. That's because balanced cable has one ground and two signal wires. Each signal wire has the same signal, but one is inverted. They both pick up almost the same interference. At the receiving end, one is inverted again and the signals are combined: the interference cancels out, and the signal is retained.
The same effect happens with pickups. A standard pickup is just a single coil around a magnet. It picks up interference, and that's audible as a hum. That hum is always there under the signal from the vibration of the guitar strings.
A "humbucker" pickup counters this by using two coils next to each other, and wired again such that the interference cancels out. However, because the coils aren't exactly in the same place, and because of complex magnetic fields, the combined signals interfere with each other a little, and it has a little less treble than a single coil pickup -- a sound that's often described as "fatter" or "rounder".