Depth Perception is affected by distance, not by focal length
OK. I hope you have read Part 1 about the incredible shrinking Space Needle and you get the idea now that long/telephoto lenses don't compress a scene, and that distance is the key player in the equation.
The closer you are to your subject the smaller the elements in the background will appear. The further away you are from your subject the larger the background elements will appear.
Let's see how that plays into the photographs of the moon and the Space Needle.
The photos on this page are for demonstration purposes. They are not actual photos taken of the Needle and moon, but of puppets in the studio. YES! THESE ARE FAKES! But they do serve to demonstrate the principals involved.
PLEASE NOTE: The "needle" and the "moon" remain stationary through all of the photos in this post–the distance between them does not change.
Getting Close
In the first pair of photos (above) the camera is positioned in very close to the cutout of the needle. In the image on the left it is fitted with a short focal length lens. On the right it is fitted with a lens double the focal length. As you can see both the needle and the moon are twice as large on the right, but relatively the same size compared to each other. The magnification increased due to the increased focal length, but the perspective remained the same because the camera, needle, and moon did not move.
Moving Back
In the second set of photos (below) the camera has been moved back a considerable distance and fitted with a long focal length lens. As above, the needle and moon have not been moved, they remain the same distance from each other. In the image on the left we see that the moon now appears much larger in comparison to the needle. The first reaction to this is often "telephoto compression."
But in the middle image the camera, needle, and moon all remain in the same positions with a short focal length lens on the camera. And we see the same large moon in relation to the needle. And in the image on the right I have cropped the middle image to match the right image to show that the same perspective exists in the photos with short and long lenses from the same camera position. The compression comes from the distance, not from the lens.
Image Quality
Something else you might notice here in the cropped image on the right is that the moon is more in focus, it is a bit sharper. The shorter focal length lens gives more depth of field than a longer lens at the same distance (but a smaller image on the sensor). Even when enlarged we can see a bit of the extra depth of field, but it comes at the expense of lower resolution because you have to crop further into the original image to get the framing that you want. Just like everything else in photography, it is a balancing act. Fill the frame and limit depth of field? Or use a shorter lens from the same position to get a bit more in focus, then crop and lose resolution.
So, repeating from above...
The closer you are to your subject (the needle) the smaller the elements (the moon) in the background will appear. The further away you are from your subject (the needle) the larger the background elements (the moon) will appear.
Back in school I was taught that you find the camera to subject distance and position (viewpoint) that gives you the relationship between the objects in the scene (the perspective) you want. And then select the focal length lens that fills the frame appropriately. My favorite Ansel Adams quote is,
"A good photograph is knowing where to stand."