I agree on the term "pistol"; it is perfectly proper to refer to any gun intended to be fired from one hand as a pistol. Only nit-pickers and purists insist on distinguishing a revolver from a pistol (in general terms) or on the use of the term "handgun". In fact, the insistance that revolvers be distinguished from pistols did not occur until the 1950's. That unnecessary distinction left no general term for a firearm intended to be held in one hand, so the purists had to redefine the term "handgun" as a blanket term for both revolvers and pistols. In the very early days (14-17 Century) a "handgun" was a gun that could be held in the hands, as opposed to a cannon which was fired from a mount or wheels.
When the self-loading pistol was invented, it was routinely called an "automatic pistol", and that term was often marked on the pistol. This was often shortened to "automatic". But again, the purists (and some law writers) made it necessary to distinguish between "automatic" and "semi-automatic". The term "self-loading" (originally a British term) has now come into use in order to avoid the politically incorrect term "semi-automatic."
As to "double action", the term correctly applies to a trigger which can both cock and release the hammer (a "double action"), as opposed to a trigger that can only release an already cocked hammer (a "single action"). So it is correct to refer to a DAO pistol as "double action", since the trigger both cocks the hammer and releases it. Strictly speaking, double action is not self-cocking. There have been self-cocking revolvers (the Civil War Starr being one of the best known) in which the trigger cocked the hammer but did not release it; the hammer was then released by pressing a separate trigger.
Jim