However, the world is filled with different types of cheating that are not called plagiarism.
Think of it this way: If we take it out of the realm of school and just go to the working world (and get publishers out of the picture, since they make it more complex but don't substantitvely change it): Let's say author A copies author B WITH author B's permission. That's not plagiarism. Indeed, no law has been broken. But it *IS* plagiarism if B did not give permission. To be plagiarism, it must be without permission.
Now, back to school ...
It is against the rules to claim work is your own when it is not, even if the original author gave you permission. In essense, you are claiming you did work that you did not do, and since the point of school is to get you to do exercises that actually teach you things, NOT doing them defeats the end goal.
I have NO doubt that somebody will try to tell you that you plagiarized, and they will be wrong, but if you claim you broke the rules, YOU will be wrong.