Profusely apologize and admit to them that what you did was wrong unless you want more garbage thrown at you by the faculty.

Unfortunately, this is how it works in the academic world. Argumentative students, regardless of how much they pay per class, regardless of how RIGHT they are, get disciplined. Splitting hairs with a professor is going to lead the professor's "gang" start closely examining you, and they'll find something wrong if they set their mind to it--even if you are 100% correct. They take offense at criticism and debating because when you question the professor's judgement, you're also criticizing the school's decision to hire the professor.

Remember that when you apply for a program, you're agreeing to abide by their code/rules. This gives them the upper hand. What students see and experience in class and in grading is about 2% of reality from the program's perspective. They are not agreeing to abide by your rules. I know this is extremely unfair, especially since you are paying them, but again the chair you sit in duringchair in classes can be filled by another paying student that got turned down for admission.