In reality, these lines are a lot blurrier. Executive producers usually are investors, but they're not necessarily the primary investors. Definitionally, they're responsible for the financing of the film. Producers on the other hand, are, in theory responsible for managing expenditure of the investment (making the film), very much like a CEO/President as you describe.
Not necessarily are producers involved in any budgeting. My First Wife and I were producers on a film that was made from a stage script of her. Our role as producers gave us vetoes on screenplay changes and gave us a hand in the production, plus royalties, but we had no hand in securing the budget or spending it.
The director is the person who actually crafts the movie. They don't necessarily have anything to do with managing the money. Sometimes you have producer-directors, but it's like a restaurant that is owned by the chef.
