Is there a name for this concept in Communication Complexity?
What you call $cc_{max}$ is known as the worst-case partition communication complexity, and what you call $cc_{min}$ is known as the best-case partition communication complexity.
These have been studied for several functions, you can find some results in the book of Kushilevitz and Nisan in chapter 7.
I'm not aware of anyone introducing the difference or the ratio as a parameter to study.