So that’s what people use in conversation? They don’t use “doctor” as a synonym for physician?
Isn’t that what they say in Britain? “GP”?
Yes
If it's as in Britain - and New Zealand is a Commonwealth country - then yes, people say, "I have an appointment with my GP."
My copies of Sir Arthur Conan Doyle's stories of Sherlock Holmes speak of Doctor Watson rather than GP Watson.
Is this a Philosopher's Stone/Sorcerer's Stone situation where someone published an Americanized edition and that's what I have?
No, I think it's because GP is a newish term.
There’s a difference between “being a doctor” and “Dr Watson”. Nobody would ever say an academic is “a doctor”.
we do use "doctor" as a synonym fro GP, but we also understand that it's a traditional customary title and they don't actually have a doctorate
As others have commented, you would indeed say, “I have an appointment with my GP.” However, people would certainly know what you meant if you said “I’m going to see the doctor.”