I wouldn't connect them.
A detective story (another name is crime fiction) is about solving a crime.
A mystery story (a category often used is mystery and suspense) is more about the supernatural and involves unusual phenomena. Of course, it could be about the police, but that doesn't really make it a detective story.
As you are Russian, I think that Bulgakov's The Master and Margarita is an excellent mystery story, although it's mainly a literary masterpiece.
How do you classify mystery and crime fiction in your country?
In my usage, mysteries often include crime, but not necessarily (for example the grandchildren spending the summer on an island will attempt to find out the source of the jewels hidden in the attic). If they are about a crime, they don't have to have a detective, because some crimes are solved by clever veterinarians or clever quilt-shop owners. And then, of course, there are the police-procedurals that I know and love. A "thriller" might not have any real mystery at all--the agent is racing to Teheran to stop another country's agent from blowing up the nuclear depot. The problem is known, just will he get there? I think most mysteries and thrillers do not include supernatural events, but maybe that's just because I don't read the last category.
The gifted actress Angela Lansbury stars in a long TV series called "Murder, She Wrote." She plays the part of a "mystery writer," writing "mystery novels." She is the one pointing the police in the right direction when the local detectives are trying to solve the murder.