I'd stop right now and get an attorney to investigate your contract with the school. Their actions at this juncture are dishonest, and one should never pay up front. The recent collapse of Silver State Helicopters in which students lost an average of 50,000 dollars each is a good example of a long line of flight schools which have done the same thing. There's likely ample precedent to support you if you want to withdraw without a penalty. The collapse at Silver State has literally shut down the flight training finance available in the US. There's very little humor left for schools which do what silver state did...and the school you describe is doing what silver state did.
Even if you had put your fingers inside the spinner to pull the airplane, the damage that's evident wouldn't have occurred...and certainly not from pushing on it, either. Given that it's nearly impossible to pull the airplane with your fingertips inside the spinner...the school's claim that you damaged their aircraft in the manner they discribed simply doesn't hold water.
The inside of the spinner will need to be examined for more information, and a close-up examination will help reveal the nature of the cracking, it's origin, and what's cracked since it first started. Clearly it's easy to spot on a preflight or post flight. Again, whether you noted after the flight or not is really irrelevant to the claim of whether you caused it. The school is going to have to come up with a far better claim than they're presently making, however, to suggest that you're responsible.
Have you looked at your training contract or rental agreement to see where your responsibilities lie?
If you're a member of AOPA and have their legal plan, this is a good time to invoke it's offerings. If you're not a member of AOPA, now is a good time to join, and invest in the minimal cost for a legal plan.
The path of the crack around the screw in the same stress pattern created by the screw, intrigues me.