Private English tutor :) And yes, the nationalities always start with a capital letter. For example: Mexican food, Italian teacher, Korean music.

To me, as a native English speaker, if you say English private tutor" it means that the tutor is from England. If you say it "private English tutor" it means a private tutor of English. If you wanted to be really clear, you could say private teacher of English. Yes, capitalized.

I use English Language Teacher. Yes, English should always be capitalized.

Hi Federico 👋 I was taught that the more the adjective is constant and describing the nature, the closer it is to the noun. As an example: if a ball is blue and big, one would say a big blue ball and not a blue big ball. This is because the nature of being blue is more certain and objective, and the nature of being big is subjective and can depend on the person describing the ball. So in your case, I guess it depends on what you want to emphasize: is the fact that this teacher is offering private lessons the main information, or that he/she is an English person? I would tend to say private English tutor, because as an English tutor, he/she can easily change from teaching privately to teaching publicly; but as a private tutor, he/she could not change so easily of language or nationality. Hope it's helpful, and not too confusing 👍🙂