It is very convincingly faked. If you look at the movie in your second link, near the beginning, look very closely at the stairs handrail. You can see how there is a “cloud of splattering rain” (almost like a cloud of smoke) around the railing. The splatering off the surface is done in that manner. How they determine where to have this splatting occur, I dont know. They could hard code it into the world, or they could examine the geometry at runtime and create the “clouds of splattering rain” whereever the dot product of the surface normal and up vector is greater than zero. And they could even scale it so that the spattering effect grows as the surface gets more horizontal. Alternatively, instead of using the up vector in the calculation, they could also use the rain velocity vector.
I tend to think that for the world, the splattering is precalculated, because you only see it in certain areas (the handrailing has it, but the steps dont). But for the player, it looks to be dynamically calculated (perhaps with the dot product method), because as his body moves, the splattering occurs in different spots.
As far as the falling rain, that is very easily done with a large poly with a mostly transparent texture. Unreal Tournament had a simple form of this in the “Dreary Outpost” board. You could see rain out the windows, but it was just 1 or 2 polys with a rain texture and the texture coords were just being offset slightly each frame (like sliding the texture along the polygon). Of course with only a few rain polys, you lose the effect if you walk through them (which is why they were places outside the window). However, give modern graphics speed, you could easily put a few hundred rain planes all at slightly different distances from the camera, some parallel, some at angles, etc. This (with a bit of care) should be quite convincing. And as the camera moves, you either move the rain planes with it, or you remove planes that go behind the camera and enable some new rain planes farther away. You can also use animated textures, so that you can make it look like the rain is being driven by gusts of wind (that video looks like it did this). Or you could also achieve that same effect by varying the number and/or transparency of the rain planes rahter than using an animated texture.
Thats my take on it
P.S. Metal Gear Solid 2 is the only reason I have even considered getting a PS2. It seems just incredible.