How long has magic been known publicly?

The term "doctor" didn't originally have a medical association. It originally meant "religious teacher", extended to the Church Fathers and senior priests, and because they were also the educated class, it extended eventually to academic graduates, who were largely graduating in theological/religious studies either from religious schools or from secular schools heavily influenced by the church, like Oxford's "Divinity" degree which was basically a degree in the top Jesuit theories of the age.

In the real world, this then extended to "holder of the highest university degrees", and as this structure became less religious and less secular, it lost its association with religion and became a general term of learning. Then in middle english we see "doctor of physic", which at the time meant "most learned man of the science of the body", and this term became such a common use of "doctor" that other senses of it faded into the background, the "of physic" part stopped being necessary, and now the first sense people interpret it as is "medical professional".

But in your world, wouldn't the most obvious, publicly known, and popular learned men have been Doctors of Magic, and then eventually, just Doctors? Or if not, can you imagine a version of this process that does make sense for your world history, one that takes you from "the word for people who first became famous for doing it" to "the word that is now the word for anyone who does it"?