Do these two types of mouli graters do the same thing?
Those two are quite different.

The first is for soft food:
The "wings" push the filling through the mesh, pureeing the food and holding back tough bits like pieces of fruit skin or seeds. It's the same principle as pushing food (chunky soup, cooked fruit...) through a sieve with a ladle or large spoon, just a bit more effective and comfortable - if you don't mind the cleaning step afterwards.
The wide funnel / splatter guard and the metal "rest" at the front show that the food mill is hung over a pot, so that you can comfortably ladle your hot food into the mill and turn the crank without having to actually lift the contraption.

The second is for hard food:
It is basically a round grater, aka drum grater. The cheese (typically use case), block of chocolate,... goes into the top channel, the grating is done by the rotating "drum". The shavings drop out at the side. It's not really obvious if you look at your photos, but the handle actually consists of two parts, connected with a hinge at the outer end. The upper part can be opened wide and the to be grated food goes under the end of the handle into the square receptacle, where it is pushed against the drum by the curved end when you hold the entire handle. Here is a random open example:

(source)  

It's a hand-held tool, that is light enough to grate directly over a pot, bowl or individual plate.

The third is simply a variety of the second. Same principle, slightly different construction.