William Hartnell was in fact the first Doctor back in 1963. In order to capitalize on the popularity of the show, it was decided to make a Doctor Who movie.
Peter Cushing was tapped to play a human scientist actually named Dr. Who in the movie. In this case, Cushing's character built a TARDIS himself from scratch.
Although the movie and its sequel were remakes of Terry Nation's first two Dalek serials "The Daleks" and "The Dalek Invasion of Earth", because of the changes to the format and premise (i.e. in the first film, Barbara is one of Dr. Who's grandchildren alongside Susan), the two movies were never considered 'canon' for the Doctor Who universe.
According to Steven Moffat, and confirmed by Doctor Who Magazine, "The Day of the Doctor" was to have confirmed that the two movies exist within the Whoniverse, as Clara was supposed to see posters for the two films hanging in UNIT's Black Archive, but the rights to simply show the two posters were too expensive.
In any event, even if Cushing's Dr. Who were to count, Hartnell still played the Doctor first so there is no error in not calling Cushing the first Doctor.