That is an incredibly bad explanation of how normal maps work.
Basically:
A height map is the actual height of the texture at that point. The highness of it. What it doesn't show is the angle and direction of that part of the texture, which means that bumpmapping algorithms need to perform crazy calculations at runtime (or load time) to get lighting angle information out of it.
With normal maps, they don't have to do that crazy math -- they just do some vector math between the light source and the normal of that pixel -- so it's both more accurate and faster.
However, you can't do "parallax" mapping without a height map. TF2 doesn't support parallax mapping, though, so it's a moot point.
There's also "self-shadowed bumpmaps", but those are basically just pre-baked optional lightmaps which are interpolated between. They're locked into an intensity, ambience, and "pitch", and their "yaw" is mostly inaccurate, and they end up being more ugly than normal maps most of time time. If we could use both at once, you could probably get something interesting out of mixing normal mapping with SSBmapping.