I really can't let this one go, the argument makes no since. The ACTION is how a firearm operates, and loads, not the way the trigger works, most triggers work the same way, you press them to the rear ( after cocking in one way or another, and there is a bang afterward, some shotguns have a release trigger, you pull it and hold it, then release when you want the shot to break, what do you call that? then there are electronic triggers, basically micro switches that operate a solenoid to hit the firing pin or directly send a charge to the primer). Bolt action, lever action, break open, falling block, rolling block, pump action, also known as a slide action, single action, double action, these all need a trigger to fire, but the trigger is not a part of the description of operation.
Single Action: Pulling the trigger does ONE thing : if the hammer has been cocked, it will release the hammer. Trigger pull is light, short, and consistant between first and subsequent shots. But does not this take 2 actions on the part of the shooter, cock the hammer and then pull the trigger?
The double action: 1 definition, pulling the trigger will cock and fire the firearm, if loaded with cartridges. As you mentioned, it cocks the hammer and fires, to me that is a single action, 1 operation to the trigger fires the cartridge. At least in a revolver.
Now in a Modern DA, revolver, as aforementioned, you can cock the hammer and pull the trigger, or just pull the trigger and it go's bang both ways. 2 ways of discharging a round.
Now DAO, why do they have the misnomer of ONLY, double action ONLY, if the the single action is not available? Since a single press to the trigger will end in a fired round. Seems to me that the other action has more than one method of operating, hmmm. Why is the other revolver called a double action? Call it a double action, when it can be cocked and fired, or pull the trigger and it indexes the cylinder and fires, but if it can only be trigger fired, it is a double action only? I will try to find the early advertisements of the double action revolver, which explains this.
Now lets confuse it some more, the DA/SA, once loaded, a single press to the trigger will be a long slow pull that cocks the hammer and lets it go, after that the SAME, press to the trigger will discharge the round from a pre-cocked hammer. This is why I prefer the term trigger cocking as applied to double action only.
I understand this is semantics, but to me are important. The term DAO, is not descriptive of how it works. To me you are saying this is the way it is, but is it not better described by my explanations.
Now, I too am an NRA instructor, personal defense, firearm safety and basic pistol marksmanship, the NRA manual does not go into this in a detailed manner as far as action types. And I teach my description of action types like I have described in this conversation.
I am also a gunsmith, and action types mean a lot to me in my profession. Clear, concise applications of which firearm I'm doing work on.