Your view of the distinction is a commonly held one but not correct. The word doctor comes from the Latin word docere, meaning to teach. The doctorate is an old degree, it dates back to the 12th century. A person with an MD, a professional degree, is a physician. It has only been recently that physicians even needed a college degree. Many of them were barbers and blood letters.
So when Obamacare assigns me to the Dr. Sweeney Todd clinic and barber shop, should I be concerned?