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Who likes a good story about etymology? Hm? No one? Good. Sit down and attend, children, as I tell you about "doctors."
"Doctor" is actually Latin; it comes from the verb "doceō," meaning "to teach." The word went to Norman French as "doctour" and to Middle English as "doctor," but the first "doctors," Latin proper, were theologians and philosophers of the Church.
Aquinas, for example, is the Doctor Angelicus, while his contemporary William of Ockham was the "Doctor Invincibilis." Because these two were more or less the rock stars of philosophy pre-Black Plague, they kicked off being called doctor as something cool.
When Europe picked itself back up after the Plague and started getting all Renaissance-y and leading to the early modern period, universities started granting "doctorate" titles to their most accomplished scholars.
Now, because the corpus of human knowledge was MUCH, MUCH smaller at this time, it was possible, expected even, for a learned man, like, say, Rene Descartes, to be proficient in philosophy, mathematics, science, law, and physick all at once.
So it wouldn't surprise you that your doctor went straight from your leeching to deliver a lecture on celestial mechanics to gathered academics. But with the coming of the late modern age and the industrial revolution, that changed.
As the entire body of human knowledge grew, so did the need for specialization. One could hardly be a polyglot mathematician engineer lawyer doctor within the span of a single lifetime, and so people started developing specialties and practices.
Through the end of the 19th century in England, to be a "doctor" meant to hold an advanced academic degree. In 1838, the Royal College of Physicians began to apply the term "Doctor" to medical practitioners, whether they held the academic degree or not.
As laws against quackery developed, it became illegal to apply the term "doctor" to those not licensed to practice medicine, and to use postnominals (e.g., Ph.D., J.D., etc.) to refer to academic degrees in formal correspondence.
American universities were likewise inconsistent with the form of address for academic doctors, calling them variously "mister" or "professor" rather than "doctor," although among academics and being addressed by students "doctor" remained possible.
Lawyers, on the other hand, in the mid-1960s, got together and decided it would be supremely confusing for us to be called "doctor," even though we had doctorates, because for those of us in private practice we could be confused for medical practitioners.
In other countries, the same confusion persists. In Japan, for example, the honorific used for professionals (professors, lawyers, medical doctors, etc.) is still -sensei, although -hakase can also be used for Ph.D.s.
Instead, Japan relies on social shaming to mock those who pretend to the title of -sensei but haven't earned it, often writing it in the symbols reserved for rendering foreign words (katakana) indicating its low status, akin to using scare-quotes in English.
Germans, on the other hand, almost never use "Doktor" academically; most of my German professors preferred to be called "Herr" oder "Frau(lein)," but "Herr Professor Doktor" or "Herr Professor" would have been more correct. "Doktor" is used for medical doctors.
This exhausts the foreign languages I know. But the point remains -- the usage of "doctor" to mean a physician of the highest education and training is relatively new (about as old as the end of the American Civil War) and we still haven't figured out how to handle it.
Saying the incoming First Lady can't use "Doctor" as a form of address is pointless bickering and dick-measuring about who has the spiffiest tertiary degree. Anyone with a "D" at some point in their academic postnomials can teach at the university level.
That's what really matters; whether you're a JD, an MD, a PhD, an Ed. D., a D. Div., or whatever, you're qualified to instruct undergraduates and maybe even some grad students. If you can do that, you deserve to be honored with an appropriate title in formal settings.
If I were to speak at a college graduation, for example, I would expect the robes I would wear would be equivalent to that institution's doctoral-level robes. If I were to teach a class on, say, con law to undergraduates, I would be referred to as "Professor Haygood," formally.
Informally, I'd ask my students to call me "Lane," but that's because I'm not pretentious. But the university provost introducing me to my peers? It's Professor Haygood, all the way.
