Fever or high temperature?

He's running a temperature = He has a temperature = He has a high temperature = He has a fever

It would be more normal in the UK to say they have a temperature. The "high* bit us implied

fever or temperature. i always hear the word temperature in my grandma's mississippi accent and associate it with old people though.

Here (Scotland, maybe even just very local) we just get 'has a temperature' the word high is implied because generally it's *high* if it's being noted. 'Adam has a temperature and needs to go home'

It’s not unusual

Yeah, high temperature, fever, pyrexia and febrile are the main ways of expressing it. But the last 3 usually mean above 38° whereas you can have a high temperature and have like 37.9°

I don’t commonly hear someone say that someone has a “(high) temperature),” but that doesn’t mean it’s wrong; I’d know immediately they’re referring to a fever. I think it might just be a regional thing. Some say “has/ running a (high) temperature,” some say “has/running a fever”.

I have a (high) temperature is, in my experience, *more* common than using the word fever. Your teachers aren't "insisting". They are simply stating a fact like any other.

Native speakers will understand both but fever is a more correct way of saying someone has a high temperature.

Yes. You can also just say "He has a temperature" and native speakers will understand that you mean "a high temperature" or "a fever."

Incredibly common and totally acceptable, in my experience much more common for people to say "has a high temperature" or "has a temperature" than "has a fever", though I'd only say "*high* temperature" if it was over 40°C.

“Has a high temperature” can be said, but it’s more common to say “has a fever”. “Has a temperature” is also used. It’s understood that it’s a high temperature, so no need to specify “high”.