Let's say that we have this dynamic game with complete but imperfect information represented in extensive form: the dynamic game
Isn't this dynamic game just identical to a regular static game? I do know that normal form games and extensive form games are convertible, but in this case, why would you even call it a dynamic game with complete but imperfect information? Isn't it just confusing to call such a game dynamic if it is a static game?
You make a good point. Let me try to spell it out for you. "Dynamic" vs. "static" are interpretive labels, whereas "complete" and "imperfect" information are formal notions.
You can interpret this game with complete and imperfect information as a dynamic game, where 1 chooses his strategy and afterward (having no knowledge of 1's move) 2 chooses hers. Or you can interpret it as a static game where they make simultaneous choices. From the point of view of game theory, because the formal description is the same, there is very little practical difference.