Please let me try to clarify, from the point of view of a RE person, a few questions raised in this thread. And sorry for the length of my post. Please enjoy anyway.
1. It is claimed that observers at different locations on earth should see different sides of the spherical moon depending on their respective location. That statement is correct. But if any imaginable distance on earth (like its diameter) is small in comparison to the distance moon-earth the effect is hardly noticeable to the naked eye. Diameter of earth 12,740 km, distance earth-moon 384,400 km (average) , ratio of 30. You can verify this effect by looking at a building head-on but from far away. Moving side-wise by a few feet will not reveal in any noticeable way one of the sides of the building particularly when the building is round.
2. We always see only one side of the moon, called the near-side. That is correct in very good approximation. The reason is that the moon is tidally locked which is a short way of saying that the moon rotates around its own axis exactly once during a complete orbit around earth. Pluto and its moon Charon are a more advanced example thereof. They both show each other only their respective near side.
3. During the course of the night the image of the moon including moon phase seems to rotate. "seems" is the keyword. It is actually you whose horizontal plane is changing its orientation with respect to the moon. Reason : earth is spherical in shape and rotates around an axis. You could clarify that to yourself in the following way :
On a piece of paper draw a nice circle and then add a vertical line through the center of the circle. This is a crude presentation of the spherical earth with the straight line representing earth's axis. Mark the north pole at the top. Now add the equator - a horizontal line again through the circle's center. Lastly mark three points at identical distance from the equator, one on the axis and the other two on the circle on opposite sides. One could say that these three points are at the same latitude. In this scenario your face is now going to represent the moon and to define its angular orientation we consider the line connecting your eyes. In the following I will call your, personal, left eye the moon's left eye. Now imagine an observer on your sketch of earth looking at the moon, the observer being situated at the point on the circle to the left of the axis. With the known direction of earth's rotation an observer at this point would see the moon rising with the "moon's" left eye well above the horizon in comparison to its right eye. Some 6 hours later this observer would be located at your mark on earth's axis. The "moon's" eyes appear now at equal height. The moon seems to have rotated !!! I leave it up to you to explore what our fictitious observer will see additional 6 hours later when the moon is setting.
4. Moon-tilt Illusion. How I hate the word illusion in this discussion. If you happen to look at the moon other than at full or new moon you see the day/night terminator on the moon's surface giving rise to what we call the moon's phases. There is no illusion nor distortion nor rotation involved in the image you are seeing. None whatsoever. You also see correctly that the line connecting the two "horns" where the terminator meets moon's perimeter is inclined most of the time with respect to your vertical. Actually, if you know enough math you can derive an equation for the tilt of this line given the position of moon and sun in the sky in terms of azimuth and elevation angle.
Trouble arises when you want to draw the line the light follows when it travels from the sun to the moon. Except for a very special circumstance drawing a straight line means you commit a serious error !!! Let me continue with a statement : If we were able to see light travel from the sun to the moon the light would follow a curved path as seen from an observer on earth. This phenomena is not restricted to the sun-moon problem but very general in nature.
So let's bring it down to earth.
Imagine you are standing on a big level field - it is pitch dark and all you can see is five very small lights which are NOT moving. Because of the total darkness you don't have any depth perception. You also have no knowledge about the actual brightness of each light; one might be actually very bright but far away, the next one very dim but close by. So, you do what astronomers do when measuring the positions of objects in space, you measure azimuth and elevation angle. Here are the numbers you measured, assigning 0 deg azimuth to the far left light and 90 deg azimuth for the far right light (you will be able to verify these numbers yourself shortly) :
Light azimuth [deg] elevation [deg]
1 0.00 15.00
2 18.43 18.72
3 45.00 20.75
4 71.57 18.72
5 90.00 15.00
According to these numbers, if you let your outstretched arm go smoothly from the far left to the far right light you would trace out a nice arc in the night sky reaching the highest point at the center light.
The big surprise comes the next morning when you revisit the field to complement what you have seen the night before. You discover the following : from where you were standing you had looked at five poles with little tiny lights on top. The center post ( light # 3) is exactly 100 meters away from you. The posts are located along a line perpendicular to the line from you to the center post. The posts are equally spaced, 25 m apart and exactly of the same height as verified with a laser beam going from the right most to the left most post just grazing all five light bulbs. Of course, if you could have seen the laser light at night it would have followed the nice arc your outstretched arm followed.
With this new information, your calculator and knowledge of trigonometry you can now not only calculate the height of the posts (quite high I might add) but also verify the numbers in the above table.
Bottom line : The path of light traveling in a straight line from the far right to the far left post (the above mentioned laser) should be drawn as an arc in the night sky. Drawing now an analogy to the sun-light-moon problem : If you climb up on one of the intermediate posts you are putting yourself in the same position of earth being located between sun and moon simulating a lunar eclipse. If you move a little bit to the side or up/down so that the light can pass you by we have the situation of a full moon. Yikes, where is the sun when you look at a full moon high in the sky ?