Math disagrees with you. You can view a three dimensional object as projection onto a 2D canvas. I suppose I should add the stipulation of 'n>2', but I'm not wrong.
It's a 2D representation(because it's flat on your screen) of what a 4D cube would look like from a 3D inhabitant's perspective.
Just as a sphere passing through a plane would like like a circle that expands and contracts yet would also be viewed as a line that expands and contracts to 'inhabitants' of this 2D flatland.
This GIF shows the tesseract passing through itself because in those sections where it must reorient, it is not passing through itself, just passing by itself in a fourth dimension, but we cannot see that fourth dimension, so this is how the 4D cube would apear in our 3D world(theoretically).
TL;DR - it's still 4D, just what 4D looks like in a 3D world.
I have all three on my comp. I agree with doimas on the fact that #s 2 and 3 were pretty bad compared to #1, but I did enjoy the twist at the end of #3
I'll explain
It's like drawing a 3D cube on a paper, now you watch a 3D cube in 2D.
That is a 4D cube, and since we live in a 3D world, we can only watch the 4D cube from a 3D perspective.