Just got a new power amp

@kijanki Right.

(We introduced a class D amp of our own design about 4 years ago FWIW.)

The question then becomes, in the context of this thread 'is it important?' WRT the thread title, it isn't. Ampex made a tape machine that was used for quite a lot of the recordings made in the so called Golden Age of stereo (1958-1963); that machine had a balanced input and output using transformers but the internal electronics was mostly single-ended. Its use of transformers allowed it to be immune to interconnect cable differences.  

One criteria of balanced output is lack of GND return current.

@kijanki So if one side is at ground, but there is no ground return current to any other part of the amp, your criteria is satisfied? Or is the fact that the speaker circuit is not complete without a ground connection a problem?

Audio Research used to ground the 4 Ohm tap of their output transformers rather than the common tap, so they could cathode cross couple the output tubes; IOW the cathodes of one half of the amp were tied to the common tap and the 16 Ohm tap was tied to the other half of the amp's cathodes, as a feedback mechanism. To do this obviously the feedback was balanced. Does this meet your criteria?

@kijanki Did you not mean to use 'S300' twice in the above post?

@kijanki It looks like that amp is. 

But:

I would not use a speaker terminal at ground as your definition, although our OTLs have both speaker terminals ungrounded and floating. 

Many ARC amplifiers do not have a grounded  'common' terminal; the impedance to ground of the common connection is the same as their 16 Ohm connection. 

IOW that is not the defining character of a fully balanced amplifier although it is a characteristic they are likely to have. 

We are experimenting with a tube amp that has a speaker terminal grounded, yet its fully balanced from input to output with dual feedback loops. 

@kijanki Its hard to know what Emotiva is talking about since there appears to be a typo or complete misunderstanding of what is happening in the opening sentence of the quote of the description; power amps do not drive the input, if you see what I'm talking about. A preamp might do that though...

Anyway, IMO/IME your definition is a bit too restrictive. Our OTLs are an example that flies in the face of it and are one of the oldest fully balanced differential amplifier designs in production. 

Fully Balanced configuration (as I call it), doubles the circuitry (cost), with little benefits (other than high output power)

This statement is false if the circuit is also differential. You do have more parts but not double. Depending on the gain of the circuit the CMRR can be quite high. 

There are amps that are fully balanced but not differential. I see no advantage to them at all, since they often offer no CMRR and do indeed have double the parts. 

Apparently Douglas Self doesn't know about our stuff since he makes the claim at the link you provided:

To my knowledge these are the first, high power, full balanced amplifiers with feedback from the input to the speaker terminal in high volume production.

We've been doing exactly that for 40 years although its been with tubes, but he might be right depending on what is meant be 'high volume production'.

Now if you want, you can build a conventional tube amplifier using a differential Voltage amplifier and transformer coupled push-pull output. In fact that is exactly how our little Gem integrated amplifier works (it only makes 5 Watts/channel, meant for headphone, bedroom or desktop use, but you could use it in a main system with high efficiency speakers). 

Depending on how you execute the ground at the output, you could have balanced feedback loops or single-ended. Yet the amp is fully differential and balanced from input to output. 

I really think you need to expand your definition, since as you have stated it so far makes a good deal of your arguments false, for example the bit 

like transformer or instrumentation amp, followed by two amplifiers - each for one leg/phase of the signal.  Speaker is connected between outputs of both amplifiers. 

This statement would only be true if the word 'amplifier' meant only a single-ended circuit. Obviously there's a bit of a contradiction were that the case since the input allows for differential circuits which are not single-ended.

Further, whether the embodiment is tube or solid state is irrelevant.

For example, I'm sure you've heard of GAS, who made the Ampzilla. That amp used an output circuit known as a Circlotron, the same as our OTLs. If that circuit were used with a differential balanced input (like we do), you'd have a fully differential balanced amplifier with only a single amplifier section driving the speaker terminals and it would be up to the designer as to whether a single or balanced feedback network would be used. 

So I really think your definition/opinion has you painted in a corner unnecessarily.  

@kijanki Our OTL tube amps are fully balanced and differential. But to the best of my knowledge, there's only one output section (which is push-pull) driving the speakers. Does this meet your definition?

If there is an audible difference, something is defective.

@kr4 I'd put it the other way 'round. We audiophiles are very used to hearing interconnect cable differences. But recording engineers, who use balanced lines in a studio, are not. When the balanced line system is set up properly (as usually seen in a studio) the 'sound' of different interconnect cables goes away. 

Having done many auditions of this difference over the last 40 years I can say that RCAs usually don't sound as good as a result and nothing is defective; this providing that the balanced line equipment actually supports the balanced standards (such as AES48). 

"Fully Balanced Amplifier" means that balanced input section (instrumentation amp or transformer) is followed by two separate amplifiers, each driving one speaker terminal.

@kijanki Could you explain what is meant by 'two separate amplifiers'? Do you mean two single-ended amplifier circuits or two push pull amplifier circuits, or something else?