Re: Take yer lazy americanisms and get ter feck!
Ah, well, the Gendarme is subtly different as are the German Federal Police (who are only responsible for the security of railway stations, ports, etc - routine Policing is strictly devolved to States because of the bad history with centralised Policing - namely the Stazi).
In France, the Gendarme are a para-military organisation distinct from the Police Nationale, who are the Police. Much like the Italian Carabinieri and Polizia di Stato.
In those countries you have the Civil Police, then the paramilitary-esque group in the middle, and then the Military.
In the case of a country which only has Police and Military, the Police are (or should be) most assuredly Civilian. Whilst that case naturally calls for a more heavily armed unit within the Police (e.g. SO19/SWAT) to fill that middle ground and support action against terrorism, serious organised crime, etc, the members of those units are nonetheless Police first, just as the SAS are no less soldier-first when they are asked to assist the Police in extreme circumstances (e.g. Iranian Embassy).