This is an interesting thread and I tend to agree with most of what has been posted.
I have seen some technical terms define differently in different industries and some definitions even change with time. Since I am an engineer and not an English major (heck English is not even my first language), I can't comment if there is one truly correct definition for what is a balanced first of second stage, but I do have my opinions.
To me the definition of a balanced system is a system that is in equilibrium and more specific to the discussion is that the forces are or will obtain an in equilibrium. From this general definition I then tend adapt the more specific definitions adopted by different industries.
In the case of the Scuba industry I tend to think that the definition of a balanced first stage is one that obtains equilibrium without being affected by the actual tank (source pressure).
The geometry of the Sherwood regulators first stages may look similar to some non- balanced regulators, but they do accomplish a constant IP independent of tank source pressure (a friend tested several Sherwood first stages and the IP was totally unaffected by tank pressure). The summation of the two sets of springs (the coil spring and the Belleville washer's spring) and the two pneumatic devices (the primary piston and the piston behind the floating volcano orifice), add up to balance all the forces and the outcome is a constant IP, independent of tank pressure.
A primary advantage of any balanced first stage is that the gas flow orifice on the primary closing valve can be larger, since the area of the opening times the tank pressure will not affect its operation. The larger pneumatic force of the larger area is balanced by an equal and opposite pneumatic force.
When referring to second stages I also tend to use the term pneumatically balanced in a similar fashion as most of the industry (specifically Scubapro).
I have equated the balancing chamber as similar to a pneumatic spring, but I said similar, not exactly a pneumatic spring. In general I tend to think of an actual pneumatic spring as one with a fixed amount of gas and therefore provide a linear (not to be confused with straight line) spring constant. Again this is just a definition that may vary in different industries, but in this situation the balancing chamber provides more of a proportional control force than a fixed spring constant. The force varies with gas pressure not position (spring forces/ spring constants are normally defined as a function of position change or compression/ extension).
One important correction I would like to make is that in a pneumatically balanced second stage when the demand valve opens the air source to the balancing chamber is not automatically cut off. The pressure drop discontinuity in a volcano orifice with soft seat type of pneumatic valve occurs around the perimeter of the volcano orifice, not the face of the orifice. Actually, depending on flow velocity, it occurs just outside the perimeter of the orifice. This is a very important concept to understand. The pressure all along the IP volume is relatively constant including the pressure in the balancing chamber. There is a very small pressure drop due to flow, but it is almost insignificant compared to the pressure discontinuity that occurs around the perimeter of the orifice where the soft seat is barely separating from it when it opens.
To fully understand how a pressure reducing gas valve works it almost requires a course in compressible gas flow and supersonic flow. Yes, the flow in that miniscule region where there is a large pressure discontinuity becomes supersonic and there is a static shock wave (a.k.a. a pressure discontinuity).
The conclusion is that the pressure discontinuity occurs around the perimeter (or just outside) of the volcano orifice (unless you retract the soft seat so far back that it is far from the flow path, the discontinuity will always occur outside the location of the smallest opening). Therefore, the gas source into the balancing chamber is not cut off, but instead is modulated by the average gas pressure inside the volcano orifice.
BTW, the term volcano orifice is not really an industry standard term, but I like to use since IMHO it is very descriptive to the geometry of most valve orifice (it does not apply to a Poseidon Cyklon 300 first stage).