"Punishment" in the Operant sense, is anything that reduces the likelihood of the behaviour being repeated. It is the opposite of "Reinforcement", which means anything that increases the likelihood of the behaviour being repeated.
I don't think you're belaboring the point; I appreciate the specialized terminology. The above makes sense to me intuitively.
BTW, "negative reinforcement" and "positive reinforcement" is something people in the US do use in every day language. I assume that training experts here also use the same scientific terminology that you are explaining. The difference is interesting. Negative reinforcement in the colloquial usage here doesn't mean here to remove something, but on the contrary, to add a negative experience, as you describe "punishment" in the scientific usage in order to dissuade behavior. Positive reinforcement, as with "reinforcement" in the scientific definition, means the same: to add a reward to encourage the behavior.
So, the following doesn't make sense to me given the above:
... "positive" means something is added, "negative" means something is taken away.
You did explain this however. "Negative reinforcement, you are removing something, i.e. taking away a stimulus when you get the desired effect, negative punishment you take away something the animal wants, which decreases the likelihood of the behaviour."
I'm not certain of the difference is between the two. Is the former to reinforce repetition of a desired behavior? For example, taking away a toy that a dog wants until it sits (desired behavior)? And would negative punishment be taking away a toy that a dog wants until it stops barking (undesired behavior)?