He did indeed tell his students that. How you translate that is down to you. His Aiki contained what some are calling 'Aiki' but was much broader. Thus Aikido.
You have your view. Keep it, use it, progress with it as I do mine.
Please provide your references and research on how you come to the conclusion that "His Aiki contained what some are calling 'Aiki' but was much broader."
While everyone can have their own views, not all views are correct. I can have a view that the air we breathe is really a mercury gas mixed with gold flakes, but that doesn't make me right.