Again correct and at the same time incorrect.
The whole of the RAF Police work from the powers that the Provost Marshall, an Air Commodore, provides them with under their Warrant of office. They also have all of the UK laws available to them as well (Common Law, Statute Law and Case Law) but only whilst in the UK or when abroad on a UK base which becomes UK sovereign territory.) There are also SAI's which form part of their powers whilst operating abroad as well.
They obtain the permission in writing you relate to, on their warrant card, issued when they pass out after their initial police training course. So no chasing round in the small hours looking for a Provost officer.
Whilst SDO in my time in the RAF I had more than a few fellow officers arrested by the Plod. The main difference is they didn't spend time in the MGR unless they were of a violent demeanour.
A Provost officer would be called out but only to oversee proceedings to ensure impartiality remained.
My brother in law, who works within the RAF legal system, has had a chortle at some of the answers on here... Keep on trying guys!!