I think you will find that Teleportation counts as movement. If you were here, and now you're there, you moved, thus you can benefit from shadow walk, etc.
Look a little closer at the description of Teleport in the PHB, and you'll see it included in movement. Teleport powers of course say 'teleport x squares' but look at the actual definition of the word teleport, i'm not sure where its defined, but its in the PHB (else how would we know what teleport x squares even means). That is where i'm confident you'll find it linked to moving.
The shadow walk ability uses the word move, because there really is only one easy way to imply going from one place to the other. They aren't trying to restrict how you move in any way (falling is certainly moving, its like flying..but downward).
(this next bit is speculation, so take it with a grain of salt)
You mention Move 2 meaning only 'move'. Well, what is 'move'? What if you're a flying creature? could you fly 2 with a move two power? Can you swim 2? Burrow 2? When can you use a special move type for moving granted by powers?
I wonder if you have a speed listed (fly, swim, burrow, climb, teleport) if you should be able to use those as forms of movement for most cases of 'Move 2' powers? I think this is poorly defined at best.