Here's how I tend to generalize the two. Mystery fiction is trying to solve a crime, usually murder, and figure out the killer. While there are undoubtedly titles out there that use criminals as the protagonists, this style of storytelling tends to focus on the good guys, the ones trying to answer the question of whodunit?
Crime fiction seems to be about criminals or ordinary people caught up in events beyond their control. Where mystery fiction ends when the killer is identified, that's often the place where crime fiction starts. Mystery can be an aspect of crime fiction, but not always. For example, a heist film has little mystery to it other than to show how the robbers pull off the deed (or not).
Perhaps I've not read broadly enough, or perhaps I'm looking for different things nowadays. I'm not sure. But it just seems that there is a chasm in the middle of this genre we call home that separates us. I wonder why that is?
I think you've hit on the key differences between mystery and crime fiction. I believe the chasm exists because, for some readers and authors, "mystery" brings to mind more intellectual characters like Holmes and Poirot and a brand of storytelling "crime" authors don't recognize as their own. I know Russel McLean hates when the term "mystery" is applied broadly.
"Crime" fiction seems to fit a postmodern world where you can't assume crimes will be solved or killers will be caught. Detectives may be too consumed by their own demons to do much sleuthing.
While I enjoy both types of story, I have a soft spot for mystery because, by definition, it intends to challenge the mind. Crime fiction's surprises are more visceral. The crime fiction I like best blurs the rational line between right and wrong. A criminal may have broken the law, but laws are man-made after all. The act in question doesn't necessarily make the the lawbreaker a "bad guy".