Helge,
A couple of things:
First, it confuses the issue by using the terms "tele" (for telephoto, I presume) to mean a long focal-length lens for a given format and "wide-angle" to mean a short focal-length lens for a given format. "Telephoto" is a lens design that allows longer focal lengths to focus at infinity with the lens physically less than its focal length from the film plane. On the other side, many short focal-length lenses for 35mm and medium formats are "retrofocus" lenses, a design that allows them to physically focus infinity on the film at a distance greater than their focal length. "Normal" lens designs focus at infinity when the center of the lens is roughly the lens' focal length from the film. All this to try to get you to use "short," "normal" (for the format), and "long" instead, just for clarity's sake.
Also, don't confuse the "distortion" introduced by short focal-length lenses with perspective control accomplished by positioning the camera back/film plane. Objects at the edges of the field of coverage with short focal-length lenses appear stretched out, and thus "distorted" due to the steep angle of the projection. Similarly, short focal lengths tend to render near objects larger than we expect in comparison to more distant objects due to the wide angle of view (this, when compared to the "normal" angle of view we get with our eyes makes images made with short focal lengths seem unreal). This has nothing to do with perspective changes that happen when the film plane is repositioned relative to the subject.
In the three images you post, you make one with the camera back/film plane/sensor parallel to the tower, one with the camera pointing up, causing the verticals to converge toward the top of the image and one with the camera pointing down, causing the verticals to converge toward the bottom of the image. This is exactly what happens when the position of the film plane/ground glass/etc. is changed relative to the subject. If you don't want convergence, the film plane must be parallel to the parallel lines in the subject.
The converging verticals toward the top of the image in your second image is what many try to avoid by using front rise on a view camera or special shift lens on smaller cameras. You may find it more "natural," but many of us don't. As you point out, the eye/brain system straightens the image out in our imagination; even when we look up at something, we somehow recognize the parallel lines. Rendering parallel vertical lines parallel on the film is a convention of architectural photography done with view cameras that has been around for years. Until recently, converging verticals in professional architectural photography were purposefully avoided.
As for the vanishing points: With the lens axis centered on the film plane, receding parallel lines will always come to a vanishing point in the center of the image. Converging horizontal and vertical lines will have their vanishing points on vertical and horizontal lines that intersect the center of the image (either within or outside of the image borders). By using rise/fall or shift, we change where the lens axis intersects the film. That point is no longer at the center of the image but displaced up, down or to one side. The positions of the vanishing points move together with the displacement.
Moving the entire camera while keeping the film parallel to parallel lines in the subject is not the same thing as using lens rise from a lower viewpoint to keep vertical lines parallel. In the first case, you are moving the viewpoint, in the latter, not. Let's say you have a six-story building. If you move the camera up to be at the height of the third floor, your viewpoint, i.e., the point in the scene that is on the lens axis of view, will be on the third floor and in the center of the image. That will be the "straight-on" point in the image and the windows and doorways at the edges of the image will be all viewed from an angle.
If you keep the camera position low, say in the middle of the ground floor, keeping the film plane parallel to vertical lines in the subject, and then use rise to get the top of the building in the image (thus keeping parallel vertical lines from converging), the point in the image that is on the lens' axis of view will be on the ground floor, which ends up being at the bottom of the image projected on the film. Everything above that point will be viewed from an angle. This effect is very different from moving the entire camera up.
You say that focal length is "very closely tied to angle of view..." Well, for any given format, the lens' focal length determines the angle of view. The angle of view for a particular focal length and film format never changes. It doesn't matter if the lens has a large or small image circle, etc. The image size (i.e., magnification) is a function of the focal length.
Note also that one can use front rise on a view camera with any focal-length lens one chooses, long, short or "normal." I often used lots of rise with a 210mm lens on 4x5 for architectural work. This is a slightly longer than normal focal length for 4x5, so the stretching of off-axis objects that happens with very short focal lengths doesn't happen at all (that's because the lens' angle of view is similar to that of our eye). In your photos, there's lots of stretching off-axis because the lens you used was a very short focal length for the format and the angle of view is much wider than that of your eye. All that stretching "distortion" is due to the wider angle of view.
Finally, to answer your question about how correcting converging verticals is done with a view camera. It's really fairly elementary. Step one, if you want the parallel lines in the subject to be parallel on the film, the film plane needs to be parallel to them. In practice in architectural photography, this means the camera back/film plane needs to be level and plumb (because we assume that the building we are photographing was constructed level and plumb). Then, the image is framed by moving the lens about with rise and/or shift. This moves the lens axis relative to the image and moves the point of view around in the image, but it keeps the parallel lines parallel. That's really all there is to it. Of course, we need a lens with a large enough image circle and a camera with enough movement to do this. Sometimes it's not possible to get what we want.
All for now
Doremus