In general, referring to yourself as a doctor or not has to do with context and situation. If you are among your educational peers or speaking to a group and you have achieved a doctoral degree as your terminal degree, then general decorum is to include the title in the introduction. If you are in a hospital and meeting a patient and you are a Doctor of Philosophy, a doctor of education, a doctor of theological studies, a doctor of physical therapy, a doctor of nursing practice, etc., then the term "doctor" can be deceptive and will cause patient confusion because patients in that setting who hear someone introduce themselves as a "doctor" expect that the person is an MD or a DO, or, in certain settings, a DDS or DPM (podiatrist). Chiropractors traditionally use the title "Doctor" in their own clinical setting because people know that they are in a chiropractor's office (chiropractors do not have hospital privileges).
I believe that Dr. Cash absolutely deserves to use the title Doctor in his own environment. If he was on an airplane and the question was asked, "is there a doctor on board?", maybe that is not the time to use his well-earned title.
The word doctor has gotten so watered down over the past two decades as every educational process has elevated their terminal degree to a doctoral level, whether deserved or not. Examples include programs that only require ~18 months of online classes done remotely with little to no classroom time, in person education, defense of a thesis, or anything remotely academically rigorous and no barriers to entry except a willingness to go $100K into debt, and everyone who starts finishes as long as they pay their tuition, and these programs result in a doctoral level degree. TCU has many such programs. Most "doctoral candidates" maintain their full time job and, other than some homework at night, and a much lighter pocketbook, and have not put in the same level of effort as the traditional PhD, which may take 5-7 years of rigorous academic research effort.
I feel that the term doctor has become so cheapened and clouded that it should be abandoned by the medical profession and the term physician should be used.