Maybe this will help...and remember that the terms DA and SA apply only to the trigger action....
Double action (DA); the trigger performs Two functions, cocking, AND releasing the hammer. This is true for both revolver and DA autopistols.
Single Action (SA); the trigger only performs ONE function, releasing the hammer (or striker, in a striker fired auto pistol).
Now, here's where it gets a little confusing, until you get to know handguns a bit better...
DA handguns also operate in two different modes, DA or SA. In a DA revolver (like a S&W, for example) you have a choice of how to fire it. Starting with the hammer down, you can choose to fire it DA mode (just pull the trigger, the hammer will cock and be released,firing the gun), OR you can fire it SA mode (where you cock the hammer, THEN pull the trigger, releasing the hammer and firing the gun).
Either way, after the gun fires, the hammer is down (NOT COCKED).
With a DA autopistol, loading the chamber leaves the hammer cocked, and the gun ready to fire in SA (single action) mode. Each time you fire, the slide will recock the hammer, leaving you ready to fire the next round, in SA mode.
DA autos also have a safety or decocker, which lowers the hammer, leaving the gun ready to fire in DA (double action -meaning the trigger cocks and drops the hammer when pulled) mode, IF you have a decocker, and ready to fire in DA mode, but on safe, if you have a hammer dropping safety.
DAO (DA Only) means there is no provision built in for single action fire. Every shot must be done DA mode, with the trigger cocking and then dropping the hammer when pulled.