"Fully Balanced Amplifier" means that balanced input section (instrumentation amp or transformer) is followed by two separate amplifiers, each driving one speaker terminal.
I understand that intuitively what you wrote seems "true," that it "feels" like it's correct. But it is completely false. You don’t know what you’re talking about. Here’s a guy who actually understands balanced circuits and who notes the myth of your claim. (I suspect you’ll reject this, just as you have the information I’ve quoted from other manufacturers. I’m including it for the benefit of others):
Balanced line equipment is often also fully differential, like our amps and preamps. A differential amplifier has two inputs, and in theory only amplifies what is different between those two inputs ... A differential circuit isn’t double the number of parts of a single-ended circuit, although it does have more parts.