One of the reasons I write "cozy" mysteries is because, when I started, I couldn't find enough of them to read. Since then I joined Sisters in Crime and discovered a lot more of them. If I'd joined sooner I might never have felt compelled to write them!
To me, a cozy means a book without gore, without murders happening "on the page", without hard core detective or police characters, without anxiety raising terror, and without graphic sex scenes. A "whodunit" that involves everyday people not involved in a professional way with crime, who happen (or snoop) their way into the action and discover a talent for ferreting out the truth. None of the discussions I've seen about what constitutes a "cozy" mystery limits them this much, but that's my view.
One of the best comments I received about What Did You Do Before Dying? (after the ones about it being a "page turner") was that, even though it is not a YA novel by any stretch, the reader felt free to hand it on to his 16 year old granddaughter without worrying about her reading something inappropriate. That may not be a criteria for a cozy mystery, but to me it comes close.