Lol... Well, we are out of school now so I'm free until we resume. So for now, I'll try to keep up with regular posting.
Dizygotic twins are just the regular type of twins we call non-identical twins. They are usually a product of different sperm and egg cells which does not mean it has to be different intercourses. Everything as monozygotic twins applies expect for that two eggs and sperm cells are involved. So no, you haven't been wrong all along. Since it's dizygotic twins, they'll have the same father. The moment they have different fathers, it ceases to be dizygotic twins, it becomes superfecundation.