Is money paid to a boss—who delays approving an employee’s request to study outside working hours—considered a bribe or a gift, if the aim is to guarantee his future approval of the study request?

What the president demands is a forbidden bribe, because he is obligated to uphold justice without recompense. If it is not your right, then he is consuming illicit gain. It is permissible for you to pay if you cannot obtain your right except by doing so, and the sin is upon him; otherwise, it is not permissible.