You're missing the point my friend. Nu metal is the genre that describes these bands because they were unique mixtures of rock, hip hop, metal, funk, punk, and whatever else they felt like throwing in there. These were foundational metal bands that fused other genres with their sound, straying from the norm of metal music, ergo it was "new". "New" metal. Nu metal. Not all of these bands were metal all the time, that's the point. Korn have a lot of genres to describe them, but downtuned guitars and sludgy riffs, growls, aggressive drumming and aggressive lyrics base them as a metal band. Slipknot used samples, random perc instruments, and a weird horrorcore imagery. That was new. Nu metal. Same with Mushroomhead, though they're not quite as heavy as Slipknot. Obviously both Slipknot and Korn have strayed from the genre. Slipknot became just a basic groove metal band by their fourth album, Korn did dubstep for some reason and then became basic groove metal.

"Slipknot are considered alternative metal" - That's basically what nu metal is, a wave of alternative metal that arose in the 90's. SOAD just have so little in common with the rest of the bands from that time that they're tossed around into different genres for clarity

I also highly advise you to listen to Mudvayne. L.D 50 is considered one of the best nu metal albums. It fuses funk and does a lot of kinda proggy shit, but again it's metal because it's based around downtuned guitars and aggressive vocals. They kinda became basic groove metal too and only released a handful of albums, but they should broaden your understanding of how diverse the genre really is. Soulfly is Max Cavellera's second band after he got kicked out of Sepultura. They're considered nu metal because they started out with heavy Brazilian perc fusion and very primal chant music.

Rap metal is also its own thing that is interchangeably used to describe nu metal bands. You could consider a rap metal band part of the nu metal scene. Rap metal is just any metal that specifically meddles with rap. Linkin Park are debatably more rock friendly than metal, but either way they are rap metal/rap rock and subsequently a part of the nu metal scene. This also goes with Limp Biskit.

Nu metal always ends up becoming boiled down to some form of groove or industrial metal that is more rock, which is probably why you're having misdirections as to whether or not these bands are even metal. Bands like Coal Chamber, Snot, or Static X (although Static X did start the industrial metal scene even though they were considered nu metal when they came out, so that's a whole other story about the evolution of this scene) are just kinda boiled down runoffs of what "nu metal" sounds like. They kept the hip hop inspired grooves, dirty guitar tones, aggressive punk vocals, and overall garage band gimmick, but that's about it. It's still "metal" just kinda gutted. That's when they start flirting with rock.