Question regarding religious names and hospitals:?

Many hospitals in the United States have religious names (not all of them). In the past I have been told that these hospitals were operated and maintained and managed by religious groups (thus the reason for the name) and that other hospitals were usually maintained by the community and some were private. Today it seems that all the hospitals are run by corporations (both of the one's in my birthtown are). With many of the Catholic hospitals the nuns or whoever are still very much a part of the hospitals (St. Mary's here still has the sisters of St. Mary and at Mercy in Oklahoma City the Sisters of Mercy are still there, but do not maintain, operate, or manage the hospital it's done by a corporation). So what about the other hospitals with names like Baptist, Presbyterian, Methodist, Lutheran, etc, etc, etc. Do the religions still have some part in those hospitals? What part/role do they have? Today, is there a difference between the 'religious' named and nonreligious named hospitals?