Take Mr Smith as an hypothetical example.
Mr Smith goes to college and trains in medicine. He graduates and becomes Doctor Smith.
Doctor Smith works hard, and trains to be a specialist in brain surgery. He becomes Mr Smith again.
Mr Smith becomes a expert in his field, teaches other doctors, and his title becomes Professor Smith.
Is the above true? What do these titles really mean, and why do some doctors not use there title when they become specialists?
It’s not true in the United States. A “Dr.” remains a “Dr.” regardless of his or her specialty. If Dr. Brown goes into teaching at a medical school, he might hold the position of “professor,” but will most likely will continued to be address as “Dr. Brown.”
Additionally, at most American colleges and universities, professors are usually addressed as “Dr.” anyway.
Your description is true in the UK, but false in the US and Canada. (I don’t know about other countries.)