That is actually the version of the theory that I do agree with but it isn't the original one as presented by the Cahiers du cinema writers. It was originally only meant to apply to directors who do NOT write their own work and work within the confines of the studio system, yet still maintain a stylistic imprint through mise-en-scene, directors like John Ford, Howard Hawks, and Alfred Hitchcock were the original auteurs not Charlie Chaplin, Ingmar Bergman, or Akira Kurosawa as they exercised too much control over their films. The auteur theory is more of a way of looking at film rather than a real theory, that's why even in the original French it's called La politique des auteurs (or auteur policy) and not an actual theory. I think that the original theory was very valuable in legitimizing the Hollywood film as art, but it isn't really applicable or accurate today. However, I still think that looking over a director's filmography is like reading all of the works of an author, it is different than looking at the filmography of a set designer, a screenwriter (most of the time), an actor (again, most of the time), a producer, a composer, etc.
I do believe in auteurism with your definition. I believe that actors like Kirk Douglas, Jackie Chan (even when he doesn't direct), and even someone like Will Smith is an auteur. I also believe that a single film can have multiple auteurs such as Taxi Driver with Scorsese, Schrader, De Niro, and Herrmann all serving as key authorial talents. Even producers like David O. Selznick on Gone With The Wind was the auteur. There are a lot of valid criticisms even to this version of the theory, but to be honest I don't find any of the alternatives to the theory particularly convincing as they usually are focused more on the social context of the film than the actual formal elements of the film(s) that can be observed. Then again, some elements of auteurism are just as ridiculous but I don't really accept those, such as reading the biography of the filmmaker to understand his work, it is generally not relevant.