I understand that this term refers to private schools only in certain regions of the UK, not the entirety.
Not all private schools are called "public school" and within this group the location within the UK is not a criteria. The criteria which determines this classification is the age of the school and the universities which they serve. The public schools aim to send a high proportion of pupils to the universities of Oxford and Cambridge in the UK. This fact is peculiar to the UK and is part of the de-facto culture of the UK system.