You are correct about the calculation for the four hour estimate, and I think the "12AH/20HR" means that the 12 Ah capacity is based on a 20-hour discharge (i.e. discharged at 12/20 A, or 600 mA). In general, you will get less than the 12 Ah out of the battery if you discharge it faster, but you will probably still get close to the four hours.
I'm happy you like the articles. I was ready for more of a discussion after Brad's post, but looks like he did not want to follow up on it. Your claim would be more credible if you were basing it on more than a Wikipedia article that starts with a disclaimer about not citing any references or sources. You acknowledge that "the terms get used kind of interchangeably these days" in the field; are you saying all those people are wrong because they do not use your more narrow definition? That could be an interesting discussion, but you would have to back it up with a lot more than "that's how I learned it".
I'm not sure what your point is with all the sine wave business; you realize that is just restating your claim and that rectangular waves (with non-50% duty cycles) can be expressed as sums of sine waves, too, right?
It would also help if you addressed the other examples I gave (test equipment, textbooks). For example, it would be interesting if you were involved with some old function generator design team at a reputable manufacturer and could tell us about an internal discussion about "square" vs. "rectangular" and you knew that the only reason the feature got called "square" was because "rectangular" couldn't fit on a button, and everyone else in the industry just followed suit. Of course, even then, you would still end up acknowledging that "square wave" is in some sense the more generic term for "rectangular wave".
I'm not sure what you are asking. Is your safety guy saying the fuse approach is not acceptable? If you're thinking about using a 12 V lead-acid battery, those are commonly used in alarm systems, and I doubt they all need to get building permits. If they do accept the fuse or other current/power-limiting device right at the battery, the capacity shouldn't really matter, and you could use a 40 Ah battery (it would be the size of a car battery) to give you 4 A for 10 hours.
That generally sounds reasonable. The current at peak loads might be higher than what you calculated, so the performance you see in the max torque sense could be lower that what you saw with the original NiCd pack (when it was new). I have no experience with the Panasonic Evolta, but if the performance ends up being limited, you might look around for NiMH cells that are better about high discharge rates.
Also, you probably should not use the original charger for the new NiMH pack.
What is your basis for the 50% duty cycle requirement? In my experience, from textbooks to discussions with engineers to labels on test equipment, "square wave" means what you are calling "rectangular waveform". For instance, I think most function generators have a "square wave" button to access the rectangular waveform-generating features, not a "rectangle wave" button.
I'm not sure what you mean by "mandatory", but in the typical, government-backed sense, that is a very bad thing. Imagine if you bought a bottle of water without knowing how much water was in there and someone put you or the seller in prison.
If you're just talking about making a capacity specification a personal prerequisite for you to consider buying a battery, keep in mind that there are all these other variables, including discharge rate, temperature, how old the battery is, how low the voltage can be before you consider it discharged, and on and on.