^^^^ I've only once had a written apology from the Police. We'd been driving a byway somewhere around the "Hogs Back" in Surrey and emerged at the tarmac road to find a Police car, a resident had apparently complained, Plod about to lift us for "driving on a footpath". I got the marked map out, showed him the route, explained the difference between path, bridle-path and Byway ( Old money, pre the current definitions and I've forgotten the old terminology...). This was news to him... he then said he'd get us for driving across the footpath to get to the road. Cue another conversation about the dropped curb and how would one get from the legal BOAT to the road... After about half an hour it sank in that we were not a bunch of Neds ripping up the countryside, that we seemed to know what we were talking about. He demanded all our names and addresses, sight of licenses etc, very officious and unwilling to concede he was talking nonsense. We left it that he'd check the legal situation and off we went.
The formal apology arrived 3 weeks later.