Well, I'm a guitar player *and* and an electrical engineer, and I can also hear the difference in battery types in certiain distortion circuits. The rumor I heard was that it was the cell chemistry (carbon-zinc vs. alkaline) that cause the difference in sound, not the brand of battery. The story was that carbon-zinc batteries (which were used in the 60's) sounded darker because they put out less voltage.
I didn't think there could be an audible difference so I went to the 99 cent store and bought a few packages of both 9V battery types. I tried them in my Tubescreamer and I was surprised that there was a real difference; the alkaline battery sounded brighter than the carbon-zinc. Next, I made some voltage vs. current measurements and found that the carbon-zinc battery actually had *more* voltage than the alkaline battery, not less. However, the voltage vs. current plot showed that the carbon-zinc battery had more voltage loss as more current was drawn. This means that the carbon-zinc battery has a higher internal impedance than the alkaline cell.
Battery internal impedance is important in a distortion circuit because it determines how stable the 9V supply will be. An ideal battery has zero internal impedace so any stray signals that try to get onto the 9V bus will be shorted to ground. If the impedance is higher, signals created in one stage of the disortion circuit can get onto the 9V bus and interact with the other stages. A signal on the 9V bus will look like a change in supply voltage to the other stages in the circuit, so an additional signal path is created within the circuit. When this kind of interaction exists, high frequencies are the first to be diminished because the 'filtering' is worse at high frequencies than at low. Anyway, my guess is that the carbon-zinc 9V batteries have less filtering ability than alkaline batteries, so signals are more free to interact within the circuit and kill highs.
I know many people will think this is all b.s., but I've found that guitar electronics are very different from the kinds of electronics that most people use. Guitarists actually want distortion to occur, and when distortion occurs, strange things happen in the circuit. Most electrical engineers don't think too much about what happens when their circuits distort because they don't allow their circuits to distort in the first place. In this respect and others, guitar electronics is its own field of study so it is common for a good hi-fi engineer to come along and tell the guitar players that everything we are doing is wrong!
Another example of this is the effect of guitar cable on sound. Hi-fi and studio electronics are designed to have a low output impedance and a relatively high input impedance. This allows long cable runs to be used without significant signal loss, so using better or worse cables isn't going to cause an audible difference. But some guitar players have noticed that some guitar cables sound different than others, and this is in conflict with what hi-fi and studio engineers believe. It turns out that guitar pickups actually have a very high output impedance. Their internal impedance is mostly inductive, on the order of 2 to 7 Henrys (not millihenries). On the cable side, the cable lengths are much longer than in a hi-fi system, at 20 feet or more. With that much cable, the cable capacitance is large (700pF or so) and this creates a resonant circuit with the guitar pickup inductance. I know that some hi-fi nuts talk about speaker cable resonance as if it made a difference (it doesn't), but in the guitar world, the resonance exists (around 3-4kHz is common). The resonance puts a peak in the guitar pickup frequency response and it *is* audible. I've experimented with adding other small capacitances to the pickup and moving the resonant peak around to create different effects.
Since guitar cable capacitance has a direct effect on the sound of the guitar, it should not be surprising that guitar players will say they can differences in guitar cable. Some cables have more capacitance than others, and some people use short cables that have much less capacitance than is typical. Recently, I was at a studio doing some direct recording and the studio engineer handed me a very short cable he had made. I tried it out with a Stratocaster and heard some harsh noises coming out of the studio monitors. So I tried one of my 20' long cables and the extra capacitance took the edge off the guitar sound. The studio engineer was very impressed and didn't seem at all disappointed that his special short cable sounded bad.
Getting back to Eric Johnson, he's a frustrating guy because he believes *everything* changes the sound of his guitar. In certain cases, some of the weird things he believes are actually true but many other times, he is wrong. One story is that he had his guitar technician remove the steel screws from his guitar speaker cabinet and replace them with brass screws, hoping for an improvement in sound. Another is that he removed the screw from the bottom of his Fuzz-Face and used rubber bands to hold the cover in place. He says removing the screw made it sound better! Personally, I think Eric Johnson is more likely to be wrong than right, but he's right often enough that I investigate his opinions for myself and make my own decisions based on the facts I discover.