My favourite horror video game series had two games released on the original Xbox: Fatal Frame (Project Zero in PAL regions) and Fatal Frame II (Project Zero II). Both are compatible with the Xbox 360 (though the PAL version of the first game is not compatible, only the NA or JPN versions).
Both have very gloomy settings, the first in an old haunted mansion, the second in an abandoned haunted village. There is very little blood in both games; most of the horror comes not from gore, but from the atmosphere (excellent use of sound effects, music, and lighting) and the "combat" system: the main gameplay of the game is fighting ghosts by taking pictures of them. There are a fair amount of surprising "jump" scenes as well, where things pop out at you, or things happen where and when you don't expect them. The game has plenty of puzzles, both specific minigame sort of puzzles to open doors, etc. and also longer mental puzzles, putting together pieces of narrative to figure out where to go and what to do.
I have no idea what "A "Run Away" Game (Like amnesia)" means; presumably Amnesia is another game title, but not one that I've played, so I don't have any comparison. However, hitting 5 out of 6 of your requirements is pretty good, I'd say. Might be worth you looking into at least. :3