Whilst I can't give a definitive answer to the question posed, I can offer some thoughts in no particular order.
For the RAF, squadron badges were a feature from the earliest days and were 'official' having been formally designed and approved by the Chester Herald (?). Almost every squadron had an approved badge and these were very often turned into a wire blazer badge and/or a woven badge worn on the flying suit. When I joined the RAF in 1963, I don't recall that there were too many non flying unit badges about, although Commands (of which there were 11) and Groups and AHQs, as well as stations, also had badges which had been approved.
With time, I noticed that 'patches' began to appear often representing an aircraft type and also denoting 'X' number of flying hours etc. By the 1970s it seemed that everybody and their dog had a badge and there seemed to be no boundary and 'authority' seemed not to have a particularly view on badges as long as they weren't too 'extreme'. Now every 'liney' has a badge.
The best unofficial badge I ever saw and which I once 'pirated' when I had to give speech was the usual outline of an official badge and the centre was a spanner crossed with a condom. The motto, predictably, was: 'If you can't fix - F@:k It'. It was designed by an SNCO wireless operator on detachment at the RAF detachment at Kuantan in Malaya - sadly I have forgotten his name.
Old Duffer