Actually according to wikipedia source definition of reinforcement of behavioral psychology and operant conditioning, punishment is a subtype of the 4 types of reinforcement:
reinforcement is a consequence applied that will strengthen an organism's future behavior whenever that behavior is preceded by a specific antecedent stimulus. This strengthening effect may be measured as a higher frequency of behavior (e.g., pulling a lever more frequently), longer duration (e.g., pulling a lever for longer periods of time), greater magnitude (e.g., pulling a lever with greater force), or shorter latency (e.g., pulling a lever more quickly following the antecedent stimulus)... There are four types of reinforcement. Positive reinforcement, negative reinforcement, extinction, and punishment. Positive reinforcement is the application of a positive reinforcer. Negative reinforcement is the practice of removing something negative from the space of the subject as a way to encourage the antecedent behavior from that subject.
Extinction involves a behavior that requires no contingent consequence. If something (good or bad) is not reinforced it should in theory disappear. Lastly, punishment is an imposition of aversive consequence upon undesired behavior. Punishment by removal is a common example or removing a benefit following poor performance... there is also negative reinforcement, which is characterized by taking away an undesirable stimulus. Changing someone's job might serve as a negative reinforcer to someone who has back problems, (e.g. changing from a labourers job to an office position).
Regarding the question in your last section, your seatbelt case should better be viewed to increase good behavior, since the default no action which is normally considered to be morally bad but not absolute, for example, in case you need to get into your car and run away to escape some immediate danger. Or your seatbelt suddenly gets broken during driving.
Thus it's better to view your case as the negative reinforcement type as your teacher claimed since you're reinforced to be rewarded by removing the universally perceived annoying but not absolutely un-bearable dinging sound as opposed to some serious punishment assumed in OP. And your case is not unlike above WP's example of changing from a labourers job to an office position with back problems.
