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.
If you ever do watch Midsomer Murders, brace yourself! While I often include "cozy-ish crime" with cozies, Midsomer is far too gruesome to qualify. The setting of Midsomer Murders is similar to the villages Agatha Christie writes about and the plots are nice and twisty, but the methods of murder are shocking.
On the other hand, I just finished reading Deborah Crombie's A BITTER FEAST, and I was thinking about whether her books could be considered, at least, cozy-ish. Some characters swear, which seems normal rather than shocking - at least to me. And while there are murders they are not usually shocking. Family is central to her books - the family evolves as the books progress, which is one of the things I love about them. Another thing I like is the realism about family responsibilities - the need for child care even when (or especially when) a murder is being investigated. Dogs are an important feature in many of the books, and the author doesn't skip over important things like feeding the dogs and taking them for walks, training young dogs, etc.
Some of the books are darker than others, but I don't find them remotely depressing.
I love the Richard Jury/Melrose Plant books, and I think I'd call them "cozyish", too.
I think the "rules" of cozies, and of mysteries in general, are kind of arbitrary. They haven't been revised for modern readers - I think today's readers can overlook a degree of cursing in mysteries. And the rules for mysteries in general caused an uproar even when they were first circulated. Here's that list:
Ronald Knox was a mystery writer in the early part of the 20th century who belonged to the Detection Club, a society peopled by such legendary mystery writers as Agatha Christie, Dorothy Sayers, G. K. Chesterson, and E. C. Bentley. Among his novels: The Viaduct Murder, Double Cross Purposes, Still Dead.
Knox was also a Catholic priest, which is perhaps why he was tempted to write a 10 Commandments of detective fiction. If you write such stories, thou shalt obey these laws:*
The criminal must be someone mentioned in the early part of the story, but must not be anyone whose thoughts the reader has been allowed to follow.
All supernatural or preternatural agencies are ruled out as a matter of course.
Not more than one secret room or passage is allowable.
No hitherto undiscovered poisons may be used, nor any appliance which will need a long scientific explanation at the end.
No Chinaman must figure in the story.
No accident must ever help the detective, nor must he ever have an unaccountable intuition which proves to be right.
The detective must not himself commit the crime.
The detective must not light on any clues which are not instantly produced for the inspection of the reader.
The stupid friend of the detective, the Watson, must not conceal any thoughts which pass through his mind; his intelligence must be slightly, but very slightly, below that of the average reader.
Twin brothers, and doubles generally, must not appear unless we have been duly prepared for them.
*In truth, most of these rules have become outdated in current mystery fiction.
Source:
https://www.writingclasses.com/toolbox/tips-masters/ronald-knox-10-commandments-of-detective-fiction