> When a camera zooms in or zooms out, there is a change in the geometry of perspective.
This is the opposite of what happens. The perspective is exactly the same, only cropped to a narrower field of view.
Changed perspective would be counteracting the lens zoom with foot zoom -- now this causes the change in perspective you talk about. But it's caused entirely by moving the camera. The zoom is just to preserve framing and could just as well be accomplished by cropping in post, assuming sufficient resolution.