Actually, there's no inside. Maybe I wasn't clear enough. It's same situation as in Pocket Cube. You have only 8 corners. In case of hypercube 16 'corners'.
In that case the puzzle you have shown should be equivalent to a 2x2x2x2. The pieces inside the sphere should behave just as some of the pieces outside the sphere and you haven't introduced anything new. At least with this particular puzzle. I do believe you've shown a way to bring out the insides of the higher order NxNxNxN puzzles. So the method used to construct this is certainly new and I believe has great value.
As multicube I understand every puzzle which has another puzzle(s) with smaller order within it and pieces of inside puzzle are made visible by adding circles on the faces. For example mentioned above crazy 4x4x4 or your Real5x5x5 Cube. As such multicubes are subset of crazy cubes which have circles on faces.
Actually its the other way around... atleast for the NxNxN puzzles. The crazy (or circle) cubes are a subset of the multicubes. The multi-NxNxN puzzle contains all the possible real pieces of that order puzzle. The circle puzzles can contain all the real pieces too but many just contain some of the real pieces particulary when you get to the 5x5x5 and greater puzzles. This isn't always true for puzzles with the non-cubic geometry as the circles can introduce virtual pieces which aren't present in the 'multi' version.
This also isn't true for the "crazy plus" puzzles. They are subsets of bandaged higher order multi-NxNxN puzzles. I discuss that here:http://twistypuzzles.com/forum/viewtopic.php?f=1&t=21467
Let's take crazy 4x4x4 II. It has additional pieces that don't have representational cubes inside. It is crazy cube, but not multi cube.
Not true. Look at this picture:
The piece which you think is new is actually a 4x4x4 Face X-center. True it's not inside but its there on the outside of a normal 4x4x4. You've just stickered different faces of the cubie.
The puzzle I've posted in animation it's also crazy but not multi, the simplest case.
Internal hypercubies starts from odder=3. Just as in 3d.
Well the normal 2x2x2x2 is a multi-2x2x2x2 if there is no inside and I believe your puzzle is equivalent. I believe if you solve the outside 2x2x2x2 then your new volumes inside the spheres will also be solved. However go to the 3x3x3x3 and the 4x4x4x4 and I do believe you'll be introducing new pieces. If the 5x5x5x5 is like the 5x5x5 you'll need 2 concentric spheres inside each (4D equivalent of 3D term "Face") to make the Multi-5x5x5x5.
P.S. I don't use "Multi-" to just mean show the pieces of the puzzle inside. I mean show all the possible volumes inside 3D space created by the cut planes apparent on the surface of the puzzle. A multi-megaminx if I were to ever use that term would contain the pieces of a master pentultimate. Those pieces however would be in the space "outside" the megaminx. To avoid that confusion I use the term face turn multidodecahedron. Another example... the Multi3x3x3 has 27 pieces and contains the 1x1x1 core cubie. Not sure everyone here actually considers the 1x1x1 a "puzzle".