Does a med spa in Kentucky need a medical director?

Short answer
The cited rules do not settle this — Kentucky names no med spa "medical director." They do establish that esthetics licenses stop at cosmetic work, and that diagnosis or treatment of any condition is practicing medicine and needs a medical license.

Kentucky's cosmetology chapter draws the med spa line itself, in one sentence. No person shall engage in the practice of cosmetology, esthetic practices or nail technology for other than cosmetic purposes, nor engage in them for the treatment of physical or mental ailments — and the chapter does not apply to persons authorized by Kentucky law to practice medicine, nursing and the other named professions who perform incidental practices of cosmetology in the normal course of their own profession. The esthetic-practices definition then contemplates the medical setting expressly: it includes providing preoperative and postoperative esthetic skin care, either referred by or supervised by a medical professional, unless the acts are performed incident to treatment of an illness or a disease. On the medical side the practice of medicine is defined as the diagnosis, treatment or correction of any and all human conditions, ailments, diseases, injuries or infirmities by any and all means, methods, devices, or instrumentalities; and employing an unlicensed person as a practitioner of medicine, or otherwise aiding, assisting or abetting the unlawful practice of medicine or any other healing art, is a ground for discipline.