One cozy "rule" I don't really agree with is that the sleuth has to be an amateur. I think it's more about the setting and atmosphere. For example, Becke, I notice that you mentioned a series featuring a police constable. One that comes to my mind is the Richard Jury series by Martha Grimes, that I consider to be cozies. Of course, those encompass two settings, and I guess Melrose's little corner of the world is the cozier of the two, and he's an amateur, but even the London bits with Jury and his crazy neighbors and colleagues are cozieish. Have only seen the series and not read the books, but Midsomer Murders definitely seem like cozies to me. Even Agatha Raisin turns pro and opens her own detective agency!
Not sure it qualifies as a cozy, although in general I consider Christie to be a cozy author, but I'm currently reading So Many Steps to Death (British title Destination Unknown), which is more in her espionage category - disappearing scientists.