Adverse selection occurs when a bad result happens in a market because buyers and sellers do not have the same amounts of information. Clearly, this is not going on in the example. People’s choice of whether to drive safely has nothing to do with the amounts of information that buyers and sellers have.
Moral hazard occurs when people engage in risky behavior because they do not believe that they will be harmed by it. For example, if banks know that the federal government will bail them out if their loans go bad, they will be more likely to issue risky loans because they will not suffer if the loans go bad.
The example of driving and seat belts falls into this category. Without seat belts (the argument goes) drivers will have to be careful because they are likely to die in an accident. With seat belts, drivers can be more reckless. They can rely on the seat belt to save them if their bad driving causes an accident.