Shouldn't this be "injective" rather than "well-defined"? Consider the top of page 6 here:
"...since different motions might place the globe in the same position think about why this group operation is well-defined..."
He is talking about the group of rotations of the globe. 

If different elements map to the same element ($ab = cd$), wouldn't it
  mean that the group operation is not injective? I see nothing wrong
  with that and do not understand why the author suggests to check for
  it.

 A: When you talk about something being well-defined, you're primarily talking about it being non-ambiguous.
The author is asking why the group of rotations is not ambiguous, even though many physical motions might get you from orientation A to orientation B. You could rotate a sphere 180 degrees or you could rotate the sphere 540 degrees, it won't affect the ending position. In terms of the rotation group, a rotation by 180 degrees would be exactly the same as a rotation by 540 degrees, because they produce the exact same final result given where you started.
The important things to focus on when you define a rotation are the stating orientation and the ending orientation. It doesn't matter what path you follow to get there, just so long as the starting position and the ending position match. That's why the definition is not ambiguous, because you treat any motion that yields the same final orientation to be the same rotation within the group.
If you're comfortable with the idea of an equivalence class, the elements in the rotation group are actually equivalence classes consistent of all motions that produce the same final globe orientation given the starting configuration.
A: The author is referring to the equivalence relation that identifies the positions of the globe with all redundant rotations. It's well defined if when you apply a rotation the initial class and the resulting class don't depend on the representatives you choose. 
