In reply to Alan Maurer.

Not all Rolex Day-Date watches fall within the "Rolex President" nickname category - there are many Day-Date references fitted with Oyster bracelets or leather straps. Similarly, there are also certain "Rolex President" watches that are not Day-Date models, such as the Lady-Datejust and Midsize Datejust models that are crafted from precious metals and fitted with President-style bracelets.
Often times, people will simply just say "Rolex President" as a way to denote that we are talking about a Rolex watch with a President bracelet, rather than giving the full name of the watch and also mentioning that it has a President style bracelet. However, you are certainly correct here - in terms of official Rolex names, there is no watch formally called the "President" (that name only officially applies to a bracelet style).