One thing and only one thing determines perspective: the distance from the subject. Period.
For more information, please see:
Is there a difference between a distant shot on a 50mm lens and a close-up shot on a 35mm lens?
Is the wide angle equivalent in the image of the crop sensor?
Can a telephoto lens have a wide field of vision?
How does the focal length change perspective?
Why is the background bigger and more blurry in one of these images?
What does it really mean that telephoto lenses "flatten" the scenes?
I think your fundamental confusion expressed in the question is due to the different ways in which the different sources you are looking at use the word perspective. They don't all mean the same thing when they use the same word.
The term perspective projection does not describe the same thing as "the perspective of the image". The term geometric projection, in my opinion, is a more precise way of saying what is meant by perspective projection. In other words, it describes how the lens projects an image of the 3D world onto a 2D medium such as a film or a sensor.
The difference is covered in more detail on this question here at Photography SE:
What is the difference between perspective distortion and barrel or pincushion distortion?
Perspective describes the distance relationships between the camera and various elements of the lens' field of vision. As long as the camera is in the same place and everything in front of the camera is in the same place, the perspective will be the same regardless of the focal length lens used. The focal length will determine the angle of view (as long as the film or the size of the sensor does not change), but this will not affect the perspective at all. If you take a picture with a very wide angle lens and crop the middle of the resulting image, it will look like a picture taken in the same place with a longer focal length lens providing a narrower field of view. your cropping. You would lose resolution by throwing away most of the pixels from the camera, but the perspective would be the same.
From the description of the (Perspective) tag here at Photography SE:
Perspective is the spatial relationship between a camera and the things the camera photographs. If two objects of the same size are in the scene and one is much closer to the camera than the other, the closest object will appear to be much larger than the distant object. If the two objects remain at the same distance from each other but the distance from the camera increases by two, then the difference in the apparent size of the two objects will decrease.
If the change in focal length does not affect the perspective of the image, what affects the perspective distortion of the image?
The only thing that affects perspective is the shooting distance and the relative distances of various objects from the scene to the camera. To change the perspective, you have to change the relative distances between the camera and various elements of the scene. Here is an example of an answer to this question:
Suppose you are 10 feet from your friend Joe and take his portrait orientation photo with a 50mm lens. Let's say there is a building 100 feet behind Joe. The building is 10 times farther from the camera than Joe, so if Joe is 6 feet and the building is 60 feet, they will appear to be the same height in your photo, as the two would occupy approximately 33º of the image. 39º angle of view of a 50mm lens on the longest dimension.
Now save 30 feet and use a 200mm lens. Your total distance from Joe is now 40 feet. Since you are using a focal length 4 times greater than the original 50 mm (50 mm X 4 = 200 mm), it will appear on the second photo at the same height as on the first. The building, however, is now 130 feet from the camera. It's only 1.3 times higher than when you first shot (100 feet X 1.3 = 130 feet), but you've increased the focal length by 4 times. Now the 60 foot tall building will appear to be about 3X the height of Joe in the image (100 feet / 130 feet = 0.77; 0.77 X 4 = 3.08). At least, that would be the case if all of this could get into the picture, but that is not the case.
Another way of looking at it is that in the first photo with the 50mm lens, the building was 10X further away than Joe (100 feet / 10 feet = 10). In the second photo with the 200mm objective, the building was only 3.25 times further away than Joe (130 feet / 40 feet = 3.25), even if the distance between Joe and the building was the same. What has changed is the relationship between the distance between the camera and Joe and the distance between the camera and the building. This is what defines the perspective: The ratio of the distances between the camera and various elements of a scene.
On the other hand, even two lenses with the same focal length can have different geometric projections. This will affect the shapes of the elements of the image, but it will not change what the camera can and cannot see from the same shooting position. If the "A" box is in front of the "B" box and hides half of the "B" box from the camera, switching from a straight lens to a fisheye lens will not change the amount of box "B" visible by the camera. . That's what the prospect is. Different geometric projections can cause straight lines to appear curved or elements at the edge of the field of view to be "stretched", but that does not change the perspective.
In the real world, there is no lens with an infinite focal length. It is a figment of the imagination that can be used in CGI. To get an orthographic projection with a real camera, you need a telecentric lens.
To get a view where the objects at the end of a three-dimensional
the subject appears the same size as the objects on the side near the
subject we have to use a telecentric type lens which will give us a
orthographic view of our subject. One of the basic requirements
of a telecentric lens is that the lens must be at least as large
diameter as subject. This tends to make them very expensive.