It's not nearly as complicated as it may seem, there's just a lot going on.
Some fun facts:
The focal length of the lens is independent of the size of the image circle projected. Lens design makes the difference. For a given focal length you can have a lens that just barely covers the film format or one that is much, much larger. With view cameras, we like lenses with lots of coverage when we need to use movements. That doesn't change the size of the image (magnification) at all.
"Wide angle" is ambiguous. It can refer to a wide angle of view (think short focal length for the film format) or it can refer to the angle of projection. Lenses with wide angles of projection have larger image circles than ones with smaller angles of projection. I like to think in terms of "short," "normal" and "long" focal lengths for a given film format and leave "wide angle" alone unless I qualify it with "angle of view" or "angle of projection."
Distortion usually refers to defects in a lens' rendering. Barrel distortions makes a square look convexly rounded, pincushion distortion makes it look concavely rounded. Chromatic abberation means different colors don't focus at the same place, making a softer image, etc. Perspective is a different thing, and converging verticals falls under that category.
Perspective we can define as the relative sizes and positions of objects projected on a two-dimensional surface. Viewpoint determines perspective. Changing the angle of the film plane relative to the subject also changes perspective; the part of the film that is moved farther from the subject will have a smaller image than before it was moved.
Now, on to rise/shift: Think of the lens as a projector and the film as a screen. With a view camera we usually have a lens that projects an image that is significantly larger than the "screen." That means, we can move the projector (or the screen) around to use the portion of the image circle we want. Cameras that don't have movements don't need lenses with large image circles. View cameras, with all their movements, can take full advantage image circles that are much larger than the film format.
Let's look at the classic scenario. We have a tall building. If we tip the whole camera to point up to get all the building in, the verticals will converge because the film plane isn't parallel to the building façade. The part of the film that's farthest from the plane of the façade will have a smaller image than the part that's closest. (Note, this isn't "distortion," it's the perspective we should expect when the film plane isn't parallel to the subject plane).
If we want the verticals to be parallel, the solution is to keep the film (camera back) parallel to the façade, but then, we don't have the top of the building on the ground glass when the camera is in "zero" position. But, remember all that extra coverage the lens has? There's lots more image to use; the top of building is there, we just have to put it on the film. So we raise the lens (the projector) to put that part of the building on the ground glass (the screen).
Note that we do need a lens that is the right focal length and has a generous enough image circle to be able to do this from any given camera position. Still, you can do this with short, normal and long focal length lenses.
What's the difference between changing camera position by moving the camera up and using front rise with a lower camera position? In the first case, the optical center of the image will be centered on the film, since the lens axis intersects the film in the center. With front rise, we're moving the lens axis, so the optical center will appear lower in the final image. The looks are very different. The same happens with shift; if you shift lens or back, the optical center of the image (perspective wise) is no longer in the center of the film.
Best,
Doremus
Lots of interesting answers here, I don't have time to comb through, and address every argument and detail in all of them, so the above will have to be a proxy for most of them.
Focal length is normally very closely tied to angle of view, but I'm fully aware that it's not as simple as it might have appeared I thought it was here (weasel words I know).
But I guess a lot of it has to do with being able to focus to infinity at given angle of view, even if the coverage happens to be there. And of course, that is linked to the film/plate format too.
The classic perspective lines would probably be a good mental tool/image in thinking about this.
With a tele the vanishing point is infinitely far off, and even if the image circle is larger (making it a slightly wider lens on a larger format), the effect of front shift would be much less dramatic than with a wider lens.
Right?
The question of whether to call geometric distortion away from the optical center actual distortion is perhaps, if not exactly a question of semantics or besides that point, then a question of agreeing on what semantic realm we are in:
Everyday language, related to how humans see, or very precise scientific language.
Since the eye is a scanner and not a camera, the distorted geometry that is also part of human vision is filtered out by the brain, the curvature of the retina, the smaller resolution of the periphery of vision and the non conscious turning of the eye towards the object of interest.
Not that it is very important, but notice how I anticipated this in my use of "" around distortion in my first use of it, talking about the phenomenon of optical geometric elongation and enlargement.
Can we agree that the moving of the optical center is using the same effects, that you would get towards the sides of a wide lens?
The doorway photo posted by Doremus is a good example here.
If the lens is marked at a given FL for a given format, it would be a wider lens with a larger format?