Hmmm...so you are looking for a 2000s movie from the horror genre, but not gore-filled or slasher? You want something more tasteful. Foreign films are 100% fine, and sci-fi/fantasy elements are a plus.
One film jumped out at me as you listed this criteria, Pan's Labyrinth.This Mexican/ Spanish-produced horror film from the creative mind of Guillermo Del Toro is a fav movie of mine. My only reservation is that you might have seen it before since it received a lot of acclaim when it was released back in 2006. However, it was still made more than 10 years ago, so if your interest in horror is more recent than you might have missed it when it first came out.
One of the reasons I think Pan's Labyrinth would appeal to you is because it has very strong fantasy elements. It features a child, trying to escape a dark home environment and while I would say this thriller fits well into the horror genre, there are fairies, fauns and ancient monsters that you have never seen before that put the darkness back into fairytales. Still the most dangerous monsters in the film are human beings. Its a masterpiece that blends make-up, CGI and animatronics into an eye-popping feast of world-building that has not aged a day. It has great acting, rich atmosphere, exciting pacing, vivid production design and is a relatively unusual, original story for a horror film set against the backdrop of the horrors of Spain's civil war. It is an intense film, features death and the sadness of war, but certainly not a slasher movie. Overall, If you like Neil Gaiman's writings as I do then I recommend this.
I think some of the best horror movies of recent years have been Spanish-language. If you have already seen Pan's Labyrinth, then instead try out the the 2013 Venezuelan indie horror film House at the End of Time. This is a much more rare film to come across so hopefully you have not seen it. You say that slow-burners arouse your interest, and this is a quiet, a-typical horror movie that is more about character and backstory. In the premise a woman convicted of murdering her family is taken to the house of the crime scene to serve out the end of her sentence under house arrest. The movie begins the way you might expect a haunted house flick to, but quickly veres off into a deeper mystery. It also touches on elements of sci-fi/ fantasy and has an interesting twist of events I think. Very good acting, very subtle film. Might be too slow for some, but since jumpscares and the like can turn you off then this moody, sadder tale might strike a cord with you.
The vibe I get from you is that you like classic, masterful horror, a good story with atmosphere and things that go bump in the night. If so you surely like Edgar Allen Poe, right? If you are open-minded to animated horror then give 2013's Extraordinary Tales a watch. It is a collection of classic Poe shorts from Fall of the House of Usher to The Pit and the Pendelum and so on, each has its own unique style of animation, some are in black and white and some are in color, all are given spooky narrations from famous horror actors like Christopher Lee, Julian Sand and Bela Lugosi, and each tale directed by a diffrerent pioneering horror director. As you know with Poe, these are tales of suspense, and often a touch of the supernatural lingers.
One film I think is very underrated is Nacho Cerd?'s The Abandoned, another 2006 film. It doesn't get much love, apart from being named as one of the 8 Films to Die For that year, but it is one of the most terrifying films that I have ever seen. An adopted woman from the United States travels to Europe to discover who her family was and learn of her lost Russian Heritage. As she journeys through the countryside she finds terror lurking in the forest and the abandoned property of her ancestors. This movie is very creepy and real, but doesn't explain all of the strange occurences and secrets that lay waiting in the fog. But if you are okay with a movie that leaves some things left to the imagination then give it a chance.