I thought a lot of parts of the law were on what would the average person think? If the average person would presume they are police officers, then something is wrong. A small 5 year old child missing from its mother will not think to see if Police is written on the back of the coat. Again, I go back to my point of those that are partially sighted, tourists - Market police on the back of a jacket is going to cause problems with those streams of people.
Ah the good old Birmingham Market 'Police.'
For whatever reason West Mids lets them get away with it, its been pointed out to them time and time again but they never seem to want to take any action against them - the official story from Birmingham council is that the name is there because of 'tradition.'
Once upon a time the Market Police actually had Constable powers but this hasnt been so since sometime in the mid 60's (or 70's - I cant remember.)