OK, it goes like this:
There is positive and negative reinforcement.
The positive means that you add something and reinforcement means that this strengthens a behavior. So an example of positive reinforcement is giving a treat when a dog recalls to you.
The negative means you take something away and reinforcement means that this strengthens the behavior. So you push the dog's bottom until he sits, at which point you remove the pressure.
Then there is positive and negative punishment.
The positive bit is when you add something to make a behavior disappear. So your dog is chewing your shoe and you shout until he lets go.
The negative is when you take something away to make behavior disappear. So if your dog ignores you when you ask for a 'sit' you do not give the treat you are holding in your hand.
Reward based training techniques like clicker use positive reinforcement and negative punishment like the example of not giving a treat to a dog who does not sit.
Positive punishment is basically aversives, things the dog does not like. Mild aversives are the use of the voice or a loud noise, but aversives can become progressively stronger such as tugs from choke and slip collars, hitting, electric shock collars, etc.
Personally I have only used mild aversives sparingly, e.g. my voice to get the dogs to stop rough playing and pay me attention, or in an emergency, e.g. water when two dogs have begun to fight in earnest (and honestly I have only used this once and seen another trainer use it once in years of being around dogs).
There is no evidence to show that positive punishment training techniques work better than positive reinforcement ones and there are studies that show risks of increasing fear and aggression in a dog trained using positive punishment training.
Giving tugs on the lead is intended to be painful, what else do you think the trainer intended by it?!!! Repeat enough times and you will find that you will have trouble getting him to go anywhere near the slip lead.