Is Mount Sinai a Christian hospital or private owned?

As we know, any hospitals start name with St., Lutheran, Methodist, Presbyterian are Christian/Catholic hospital. But what about Mount Sinai or NYU Langone Medical Center or Tisch Hospital.

I went to the City Government college before, I can negotiate the tuition with the college because I am citizen,  but in the private school, tuition is impossible negotiable. 

Based on this idea, If I go to see the dentist, do you think going to the government or religious own hospital will be cheaper because the cost is negotiable, the private hospital is not?