In a single action, the trigger releases a cocked hammer or striker. It performs only that one function--a SINGLE ACTION.
In a double action (sometimes called a trigger cocking action or trigger cocker) a single pull of the trigger will both cock the firearm and drop the hammer. The trigger performs TWO functions--cocking and firing--or a DOUBLE ACTION.
In most cases, a double action gun can also function identically to a single action gun by manually cocking the hammer. This type of gun is sometimes called a DA/SA when it is an automatic, but it is understood that DA when applied to a revolver does not rule out the capability to cock the hammer and fire SA.
Then, someone got the bright idea that it might be beneficial to have a double-action gun that DIDN'T have the SA capability. The trigger cocks and fires the gun with a single pull, but the hammer can't be cocked for SA functionality. These guns are called DAO, and it was when they became common that you began to see the DA/SA designation on DA semi-autos to clarify the difference.
With revolvers, it's fairly simple. You either cock the hammer or pull the trigger, or both. For the most part, it's your fingers doing all the work. On an auto it gets a bit more complicated. The slide can also do work.
In an SA auto, the hammer must be manually cocked for the first shot (either directly or by manually operating the slide), but after the first shot, the slide action cocks the hammer automatically.
In a DA (DA/SA) auto, the hammer can be cocked manually as with the SA, but it can also be fired from the hammer down position by merely pulling the trigger.
NOW, what about DAO. Well, the slide doesn't cock the hammer, and the hammer can't be manually cocked--the only way to fire the gun is by pulling the trigger.
But the auto can be made a bit more complicated. Suppose the slide action PARTIALLY cocks the action. Now the trigger acts like a DA trigger in some ways--it must finish cocking the action and then release the hammer. But it functions like an SA in another way. Without slide action to perform the partial cocking, it can't do anything at all. That's the Glock action in a nutshell. The trigger does SOME cocking and also releases the striker. BUT, it can't do ANYTHING unless the slide has pre-tensioned the striker already.
Glock called it a SAFE-Action since it clearly doesn't function like a typical DAO action as you pointed out. However, BATF classified it as DAO for their purposes. If you handed the gun to someone and asked them to classify it based ONLY on the way the trigger functions (no fair peeking at the internals), they would have to classify it as SA--the trigger only functions after the slide has been manually operated--just like an SA with a concealed hammer would work. For example, the Springfield XD is actually a true SA (the slide fully cocks the striker) even though from the outside it's impossible to tell that it's any different from the Glock.
For some more twists, you can also have the slide reset the trigger mechanism WITHOUT performing any cocking function at all. In that case you would have a true DAO without a second strike feature. OR, you can also design a gun to make the slide tension a spring while the hammer appears to stay uncocked. The trigger then cocks the hammer, but without having to tension a spring--that gives a very light pull--might feel like a long SA pull-- with what appears from the outside to be a DAO.
For some reason, in spite of the obvious differences between a typical DAO and the Glock action, there are those (BATF for one
) who insist on calling it a DAO. I suppose there's no harm in it as long as one understands what's going on inside the pistol.