It does fit into psychological horror/thriller.

In terms of being recognised by the academy for best picture, only six have been nominated:

The Exorcist, Jaws, The Silence of the Lambs, The Sixth Sense, Black Swan, Get Out.

The other 5 did win awards at the academy as well, though (rightfully so). Out of those six, The Silence of the Lambs was the only winner and dominated winning 5 Oscars in 1992. So that's the prime example.

Still is a pitch-perfect film. The origin of how Demme picked Goodbye Horses (a melancholic banger of a song) juxtaposed to how it was incorporated in the film where Ted Levine acts in one of many scenes that makes the skin crawl is pretty amazing.

The Silence of the Lambs is the prime example given it swept the awards at the academy:

Best Picture, Best Actor in a Leading Role, Best Actress in a Leading Role, Best Director, Best Screenplay.

Isn't there a shot where a character sneaks into the room in the background?

True, that I think about it more so. I was thinking of obvious examples, but saw that OP wanted obscure examples. I better put my thinking cap on and think of more subtle ones.

There's plenty of shots in Halloween (1978) where Michael is in the background of a shot and his mask illuminates his presence.

*Rosemary's Baby and Jacob's ladder have subtle background compositing of images if memory serves me correctly.

Those "first time watching" cringefest of videos are so painful to watch. I stomached only one and it had the most forced reactions possible at critical points of the film or just gave Jack Torrance esq stares.