I agree particularly with your last sentence, Robert, and would be interested in feedback from you about how I go about adding the time element in my comment "My solution" shortly before yours (in space that is, but a year in time).
The words you put in the barber's mouth "I shave anyone who does not shave himself, and noone else" also clarify the matter. He clearly isn't talking about what he does or doesn't do to his own bristles, that subject is irrelevant. Similarly with Russell's famous set. It may include any set that doesn't include itself and none other, but that has no implications to whether it includes itself.
In fact this point still stands even if the barber just says "I shave anyone." since he's still not talking about shaving himself. Even if he has no shop or special working hours, and even if he isn't a professional barber. It's really just a feature of how we use verbs (or verb phrases) that take an object. Consider the absurd syllogism "John shook hands with everyone in the room. John was in the room. Therefore John shook hands with himself". Same goes for "shave".