One thing and one thing only determines perspective: Subject distance. Period.
For more please see:
Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens?
Does wide angle equivalent in crop sensor skew image?
Can a telephoto lens have a wide field of view?
How does focal length change perspective?
Why is the background bigger and blurrier in one of these images?
What does it really mean that telephoto lenses "flatten" scenes?
I think your basic confusion expressed in the question is due to the different ways the different sources you're looking at use the word perspective. They're not all meaning the same thing when they use the same word.
The term perspective projection isn't describing the same thing as "the perspective of the image." The term geometric projection, in my opinion, is a more accurate way of saying what is meant by perspective projection. That is, it describes how the lens projects an image of the 3D world onto a 2D medium such as film or a sensor.
The difference is covered in more detail at 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 items in the lens' field of view. As long as the camera is in the same spot and everything in front of the camera is in the same spot, then the perspective will be the same regardless of the focal length lens that is used. The focal length will determine the angle of view (as long as the film or sensor size does not change), but it won't affect the perspective at all. If you take a shot with very wide angle lens and cropped the middle of the resulting image, it would look the same as a shot taken from the very same spot with a longer focal length lens giving the narrower field of view of your crop. You'd lose resolution due to throwing away most of the camera's pixels, but the perspective would be the same.
From the description for the [Perspective] tag here at Photography SE:
Perspective is the spatial relationship between a camera and the things the camera is photographing. If two objects of the same size are in the scene and one is much closer to the camera than the other the closer object will appear to be much larger than the further object. If the two objects remain the same distance from each other but the camera distance is increased from both then the apparent size difference of the two objects will decrease.
If changing the focal length doesn't affect the perspective of the image, what affects the perspective distortion of the image?
The only thing that affects perspective is shooting distance, and the relative distances of various objects in the scene to the camera. In order to change the perspective one must change the relative distances between the camera and various things in the scene. Here's an example from an answer to this question:
Let's assume you are 10 feet away from your friend Joe and take his picture in portrait orientation with a 50mm lens. Say there is a building 100 feet behind Joe. The building is 10X the distance from the camera as Joe is, so if Joe is 6 feet tall and the building is 60 feet tall they will appear to be the same height in your photo, because both would occupy about 33º of the 40º angle of view of a 50mm lens along the longer dimension.
Now back up 30 feet and use a 200mm lens. Your total distance from Joe is now 40 feet. Since you are using a focal length that is 4X the original 50mm (50mm X 4 = 200mm), he will appear the same height in the second photo as he did in the first. The building, on the other hand, is now 130 feet from the camera. That is only 1.3X as far as it was in the first shot (100ft X 1.3 = 130ft), but you have increased the focal length by 4X. Now the 60 foot tall building will appear to be roughly 3X the height of Joe in the picture (100ft / 130ft = 0.77; 0.77 X 4 = 3.08). At least it would if all of it could fit in the picture, but it can't.
Another way to look at it is that in the first photo with the 50mm lens, the building was 10X further away than Joe was (100ft / 10ft = 10). In the second photo with the 200mm lens, the building was only 3.25X further away than Joe was (130ft / 40ft = 3.25), even though the distance between Joe and the building was the same. What changed was the ratio of the distance from the camera to Joe and the distance of the camera to the building. That is what defines 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 items in the image, but it won't change what the camera can and can't see from the same shooting position. If Box "A" is in front of box "B" and hiding half of box "B" from the camera, changing from a rectilinear lens to a fisheye lens will not change how much of box "B" can be seen by the camera. That's what perspective is. The different geometric projection may cause straight lines to look curved, or items at the edge of the field of view to be "stretched", but it doesn't change perspective.
In the real world, there is no such thing as a lens with a focal length of infinity. That's a figment of the imagination that can be employed in CGI. To get orthographic projection with an actual camera, one needs a telecentric lens.
To get a view where objects on the far end of a three dimensional
subject appear the same size as objects on the near side of the
subject, we must use a telecentric type lens that will give us an
orthographic view of our subject. One of the basic requirements
of a telecentric lens is that the lens must be at least as large in
diameter as the subject. That tends to make them very expensive.