Think of it as the "HAMMER COCK" rather than the action of the slide. Both require initial chambering of the round by racking the slide. Once chambered, the SA gun requires cocking the hammer, either by hand or by the action of the slide. The dbl action gun allows the trigger to cock the hammer, just like a typical dbl action revolver.
So the SA pistol only has one ACTION of the hammer, it falls toward the round.
The DA pistol has a DOUBLE action or movement in order to fire. It first moves rearward (cocks) then the 2nd action is to fall forward, toward the round.
A dbl action ONLY gun goes through both actions every time it's fired. A DA/SA gun typically fires the first round DA, then subsequent rounds are fired SA, the action of the slide cocks the hammer. When you're finished firing a string, you release the hammer and allow it to return to the forward, lowered, DA state. You releaze that hammer either manually (thumb on hammer, pull trigger) like most 1911's, or use a de-cocker lever, like most Baretta M92's.