Who administers orthodontic treatment: a dentist, a cosmetic dentist, or an orthodontist?

Legally. Legally, any Dentist can do Orthodontic Treatment, just as any Physician can deliver a baby. Orthodontic Specialists have at least 4,000 more hours of Orthodontic Training, and do nothing but Orthodontic Treatment. Studies have shown Orthodontists do it faster and better. And by the way, Cosmetic Dentistry is an honorific, not a specialty.
Ortho Treatment. Great question. I normally recommend that an orthodontic specialist, trained exclusively for 2-3 addtional years is best suited to treat orthodontic cases. Many general dentists do perform orthodontic treatment, but have not had the formal 2-3 additional years of orthodontic training. Like in the medical field, specialist are better suited.
Orthodontist. They are specially trained in the art and nuances of orthodontics.
Orthodontist. An orthodontist is a specialist who administers orthodontic care.