The post does not, in fact, talk about positive and negative reinforcement. It discusses, as it states, "punishment (negative) and correction (positive)". In the world of behavioral studies -- where the terms "positive enforcement" and "negative enforcement" come from, these terms have a specific meaning that is often misunderstood. To clarify:
"Positive reinforcement" means that you apply some sort of stimulus to mark the desired response and encourage it to re-occur. For example, you want your horse to touch a "scary" umbrella. You do nothing other than wait until he gets brave enough to touch it on his own, and when he does, you immediately give him a treat. He associates getting the treat with having touched the umbrella, so he touches it again and gets another treat. Soon, he is touching the umbrella freely and frequently, thus having learned what you wanted him to do through positive reinforcement.
"Negative reinforcement" means that you remove some sort of stimulus to mark the desired response and encourage it to re-occur. For example. You want your horse to move forwards when you give a light cue with your legs. You apply a stimulus (squeezing, bumping, spurs, whatever) until he starts to move forward, and as soon as he does, you immediately stop applying the stimulus. Pretty soon, he is moving forward from a light cue, having learned what you want him to do through negative reinforcement.
Both positive and negative reinforcement are designed to encourage a behavior to re-occur.
So, you see, "negative reinforcement" does NOT mean "punishment". When we talk about punishment, we would technically not be talking about "reinforcement" at all, because we are not trying to reinforce a desired behavior, but rather trying to discourage an undesirable one -- through the application of a stimulus such as pain, which is technically a "positive" application (meaning an addition) of a stimulus.
I hope this information will clear up any misunderstandings the staff and others have in regards to these terms.