Balanced Line - Wiki
"Many explanations of balanced lines assume symmetric signals (i.e. signals equal in magnitude but of opposite polarity) but this can lead to confusion of the two concepts—
signal symmetry and balanced lines are quite independent of each other." -G. Ballou,
Handbook for Sound Engineers, Fifth Edition, Taylor & Francis, 2015, p. 1267–1268.
You are certainly correct, as the two terms are used interchangeably, but, in fact, are not interchangeable.
"Signal symmetry", using a microphone as an example, is achieved using a Common-Mode Rejection amplifier.
View attachment 149123
This circuit is a Common-Mode Rejection amplifier.
A1 & A2 are non-inverting op-amps where A (gain) = (1+Rf/Ri). Each op-amp has a non-inverting gain of 2. (In this case, we will not include Rg (amplifier gain resistor).
Input impedance is effectively infinite, output impedance is effectively = 0. (Ideal op-amp)
Hot and Cold inputs are 180o out of phase.
Noise present on both the Hot and Cold inputs is equal amplitude and equal phase. (This is the Common-Mode noise to be eliminated)
Vin = 1V RMS, 1kHz sine.
A1(+) input receives a negative going 1V sine wave.
A2(+) input received a positive going 1V sine wave.
At a gain of 2, the output of A2 will be a 1V positive going sine wave, minus any noise presented at the inputs of A1. (As the output of the op-amp wants to make both inputs equal.
That said, the output impedance of this circuit is essentially zero. To BALANCE the impedance, from output to input another component is necessary. i.e. a transformer.
At the end of the day, a TRS/XLR input, into this circuit will present a noise-free signal to an impedance matching transformer to the input