O.K. Les,
I will try to answer a lot of questions with one post.
Regarding the officialness of coins and how to tell. In many cases but certainly not all, coins will be numbered to the recipient (the bottom coin on the right of my post with many coins) I believe the unit holds the roster of whom has received a coin. As far as e-bay is concerned, yes there are many commemoratives, but most soldiers don't care about these. We value those presented to us by Command Sergeant Majors or Colonels and Generals.
As for the FBI coin, I received it from the Baghdad Station Chief as a form of recognition for services rendered to his teams in OIF 1. I was part of an official ceremony where I received the coin, a patch and some other items, much like a change of station ceremony or a retirement. Basically, they were honoring me and a few others for the assistance we provided them during missions. While certainly not a medal, I doubt you will find this coin anywhere else and I value it quite highly. Many times factors such as cross service missions or missions involving civilian agencies prevent or seriously impede the recognition of good deeds. So, coins can often fill the void and ensure that the soldier is recognized while awaiting the possibility of a higher form of recognition.
In regards to Military Police vs Civilian Police, this is tricky. As an M.P. I had jurisdiction over military personnel in civilian environments. I patrolled the local towns where bases were located and worked closely with civilian authorities. If there was an incident involving soldiers and civilians, I would be tasked to resolve that situation on both ends and detain the civilians if necessary. So I would maintain that we are police officers as well. Perhaps it is different in the U.K. but in the U.S. we work with many civilian agencies, locally, state and federally.
I think Mervyn really has the right idea with this thread as the coins are interesting and there are many types to be seen. I posted some of my non police coins so Mervyn could see the styles and types available. I would also mention that one of the coins I posted is from my State Governor, it is given to military, legislative, police, fire, and others who the Governor chooses to recognize. So, could it be police, civil, military??? I guess Nick moving the topic is best for all.
Hope this answers some of the questions.
Scott.