What is the professional name for someone who is a physician of sports medicine?

Not just one. To be a sports medicine physician, a physician must complete a 1 year fellowship after residency in sports medicine. Doctors who trained in family practice, physiatry (physical medicine and rehabilitation), emergency medicine, pediatrics, and internal medicine are all elligible to sit for the sports medicine board exam. Orthopedic surgeons can do a fellowship and sit for a surgical sport exam.
Usually physiatrist. Sports medicine doctors are technically termed physiatrists but many orthopedic surgeons also take care of such patients or become team physicians.
Nothing fancy. According to the american medical society for sports medicine, physicians who practice sports medicine are usually sub specialist from other existing specialties. This means they may have completed a fellowship after their specialty in sports medicine. This is true for internal med, physical med, peds, family, orthopedics, etc. Also, they tend to belong to sports med societies.
Doctor. There are two types of sports medicine physicians. The first is primary care sports medicine physician. The second is an orthopedic sports medicine physician. The prior diagnosis and treats injuries. The latter does as well but focuses more on surgical management. Both perform minor procedures such as injections.
Sports Medicine. Doctor (md or do). Usually an orthopedic surgeon, but may also be rehab specialist, or other doctor with advanced training/interest in sports medicine.
Great question. There are many terms to describe these physicians: sports physician, sport medicine physician (prob the most used), team physician. You can recycle the same names with doctor (or doc) instead of physician. Most sports docs have additional training in sports med include orthopedists, fp, pediatrics, and im. Rarely, radiologists can avail themselves of sports medicine opportunities.