Unfortunately, this is a common misconception even among professionals who should know better. It's not your fault.. there's dozens of web sites that define it this way.
Sorry, that's wrong. ALL electrical circuits have 2 sides.. positive/negative, high/low, source/return, call it what you want. A single ended (non balanced) line just has one of the two wires at ground potential. In a balanced line, both wires are isolated from ground and have the SAME IMPEDANCE to ground. That's what makes it "balanced".
The IEEE Handbook defines a balanced system as one in which all legs of the circuit have the same impedance to some common reference point, usually ground.
It's not about phase. If you take a resistor and connect it from one of the 2 wires to ground, the impedances are different and the system is no longer balanced. The phase doesn't change. But the noise picked up by both wires will be different, and won't be cancelled at the receiving end. That's how a balanced line works.. balanced impedances.