A doctor can refer to someone with a PhD (doctorate) or MD (medical degree). A PhD refers to someone with a PhD, which is a research degree and the highest degree you can get in any field (but does not qualify you to be a medical doctor). A professor has been hired by a university or college to teach, and most of them have a PhD in their field.

Note that doctor means ‘learned teacher’ in Latin, and we only started calling medical doctors ‘doctor’ about a hundred years ago, when the title was extended to the medical profession as a courtesy (although they do not have a PhD and thus have not done original research in their field). Even though most people take ‘doctor’ to mean ‘medical doctor’, that is not the original meaning, and PhD is still a higher degree.
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"Dr." is the title worn by people who have completed any Doctorate degree (MD, PhD, etc.)

"PhD" stands for Doctorate of Philosophy. It’s the doctorate degree most commonly awarded for subjects in the Humanities and Social Sciences. (ex. PhD in Russian Literature, or PhD in Urban Geography).

"Prof" is short for professor. A professor is just someone who teaches at a college or university. Professors usually have a PhD (or another Doctorate degree, whatever would be appropriate for their field), but not always.
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Dr. is a courtesy title just like Mr. or Mrs. that is appropriate for all people who have earned a doctoral degree.

PhD after a name is called a postnomial. It’s exactly like using RN, MD, CPA, etc… and indicates a professional qualification or earned degree.

Professor is a job title. It’s the same as referring to a police officer as "officer" or "trooper" or calling a football coach "coach". Other examples are Reverend, "father", Sergeant, General, etc… - all related to a specific job. It’s a general courtesy.

The correct form is to not use two courtesy titles at the same time with very few exceptions. "Reverend Doctor" is appropriate and the military often uses "Colonel (Dr.)" for some titles. We don’t use "Professor Doctor" as a courtesy title and never use "Mr. Professor"

Also, when using a courtesy title (Mr., Mrs., Dr., Prof., Coach, etc…) we don’t append a postnomial to the same signature line. We can have each of the three examples you use in your question for the same person but we’d not use "Dr. Prof. John Smith, PhD" as this would be redundant at best.

We use a courtesy title in forms of address (such as in speaking or addressing):
Dr. and Mrs. John Smith
Mr. and Prof. Jane Smith
Dear Dr. Smith
Good afternoon Prof. Smith

We use a postnomial as a signature line or in biographics that need reference to authority. We also use the postnomial on nametags and business cards when that information is important.

John Smith, PhD
Jane Smith, RN, MSN
John Smith, CPA, DBA

Some people contend that, absent a compelling reason to do so, this practice may be seen as pretentious. Especially in social interactions. Some though are offered as "fair warning" in social interactions. For example: if introducing your local minister to a group a frat friends it’s not a bad idea to introduce using the title, "guys, this is Rev. Smith" and the same could be said of faculty being introduced to students that may not be aware of the position. In military circles it is considered poor form to dispense with rank when making introductions because it places the parties interacting in an unfair position.

It’s very common in the health professions to differentiate between levels of provider and in academia to establish qualification of the faculty. A syllabus or professor’s bio will almost always identify the highest degree in postnomial form, especially in the case of an earned doctorate.

Whether an individual prefers to be referred to as Dr. Smith or Prof Smith is a matter of individual preference and institutional tradition. All professors aren’t Dr. and all Drs. aren’t Professor.
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