These are both the same thing, just viewed from different angles

Not exactly. The balanced cable pair is just a loop closed by transformers at both ends, with the pair twisting intended to induce opposite currents which will cancel each other (convert to heat at whatever part of the loop). The signal you send, is the same over both cables at the same time, relative to ground, so it isn't "a signal + the opposite", the interferences are what becomes "interference + the opposite".

"phantom power" [...] "bias power"

Following with the loop from above, "phantom power" is 11-48V power applied by the receiver to the whole loop relative to ground, which gets used to power whatever circuitry is in the mic, and the signal gets sent back as a variation in that.

"Bias power" on the other hand, is a 3 wire connection: ground, a 5-9V positive, and the signal coming back through the third wire.

It's always best to double-check whatever data sheets, but the general difference is that if you hook up a 5V "bias powered" mic to a 48V "phantom power" system, suddenly you get: ground, a wire at +48V, and another wire at +48V. The circuitry prepared for 5-9V, is probably not going to like that.

That's why "bias powered" mics tend to use 3.5 TRS, some come with adapters to 6.35 TRS or XLR that have decoupling capacitors inside, and warn against using simple "dumb" 3.5 to 6.35 adapters or hooking them to a "phantom power" system (... although some systems can automatically disable phantom power when they detect a bias powered mic hooked up, but others don't).

claim someone is wrong for claiming they are doing balanced signals + "bias power"

Not necessarily wrong... but it would raise an alarm 🚨 and I'd ask for clarification on what exactly are they doing, just in case they let some magic smoke out.