UVW corresponds to XYZ, because XYZ corresponds to world coordinates. Since XYZ is already used to indicate world coordinates, UV is used to indicate the X and Y coordinates (2D) of a flat map. The extrapolation of W is Z in XYZ.
UVW is a more complex 2d representation, which is essentially the shell of an object that has been "expanded" from a 3D object. Imagine a box “unfolded”. You now have a flat UVW card that you can apply to your content and then wrap it back in a six-sided box without distortion. In short, the UVW map knows where you can flip the x, y, and z points to change the field.
Now imagine that the sphere is “expanded.” You might end up with something like a Mercator projection. The problem is that with this problem, when you wrap this 2d representation back into the sphere, you will get some distortion.
The term UV mapping is very often used. I do not hear the term UVW as often as described above.
The term procedural matching may be misleading. Simply put, this means that the computer follows certain algorithms to draw a realistic representation of the material, such as a tree, on the object, giving the impression that the grain moves completely through the forest so that it can be seen on both sides of the object. Procedural matching can use images or not or a combination of approaches ... it all depends on the "procedure".
Finally, there is no need to first convert the “three-dimensional procedural texture” to “UVW”, since UVW and XYZ mean virtually the same thing: they either refer to the world, or an expanded image of an object in the world, or, if you like, a “piece” of the world, as in the sky. The fact is that UV or UVW refers to the display of images / textures.