Glad to be of help. There are so many ways to get your head around this that it's really a case of "the more, the merrier" as far as explanations go.
AgX,
The argument about how "perspective" is defined goes on and on here. I prefer not to get too involved in that, rather just make sure I qualify my terms enough so that it's clear what I'm trying to communicate.
Really, though, it is good to distinguish between the relation of lens position relative to the subject and the position of the film relative to the cone of light the lens projects. The lens-to-subject distance really determines camera position, with the film just tagging along at the right distance for focus. Any applied movements change the film position in the cone of light that the lens is projecting. The "standard" position is centered in the cone and perpendicular to a line from the center of the lens to the center of the film. That gives us the default lens projection and is the norm for fixed-body cameras. View cameras enable us to deviate from this norm and thereby change the projection and the relative sizes of different parts of the subject from "normal." What we choose to call this is really irrelevant as long as we understand the principle and apply it correctly. I have no qualms using "perspective" in discussing this, but sticklers like to reserve "perspective" for the relationship of subject elements from the lens position. That leaves no commonly-used term to describe the changes in subject sizes on the film when applying movements. I'll continue to use perspective for both.
Precisely. I'll emphasize that it is the difference in the ratio of the distances from the center to the edge of the image circle that determines how much relative enlargement is introduced. Really, just a function of the projection.
Well... since it's the ratio of distances between nearest and farthest lens-to-film distance, yes, anytime you tilt or swing the back, you're going to increase the distance on one end/side and reduce it on the other, introducing enlargement/diminution. Yes, this is somewhat independent of the angles of the lens projection, but still, it's altering the projection in much the same way with the same general effect.
Note that using rise-fall/shift moves the film closer to the edge of the image circle and changes the relation of lens-to-film distances. It is precisely the enlargement of one end of the film in relation to the other that "corrects" converging verticals. While rise-fall/shift are easy, they are just one way to get a desired relationship between lens and film. Point-and-swing/tilt will do exactly the same thing. Using both techniques together results in more effective rise/shift.
Again, it's really just the relative position of lens and film that matters. I often think of where I want the lens to be relative to the film in space and figure out from that which movements I can most simply apply to get that relationship. There are often many ways to achieve the very same lens position to film position relationship.
Best,
Doremus