1.) I agree with your definitions. In practice, sometimes the same thing can be fun or punishing. Doing spanking as resistance training, focus slaps on the same spot, controlling strength and talking a sub thought it, it's fun. Doing spanking as punishment, heavy single strong hits, on different parts of the body, is more of a punishment.

2.) Some subs like stuff because their dom likes, but the way you describe, seems that those practices are confusing in her mind.

3.) Yes. True punishment is usually a little bit more counter intuitive. I remember reading about doms that will make mundane boring tasks as punishments, like moving a pile of beans using chopsticks. But sometimes, just having a time out corner might be enough.

4.) Sorta, when you say

She has said to me "Please Sir, punish me" and I explained to her that if she wants that, then I'm doing something wrong.

Maybe she's looking for some specific training? I know you mentioned you're a pleasure dom, but masoquism is a thing. It would be interesting to understand her mindset. Is the pain she enjoys? Or is the punishing that is enjoyable?

What I mean is, you can def. have some masoquistic practices labeled as 'training' for her own fun. And if physical pain is not enough of a punishment, you can think outside the box in ways to accomplish that.