I should have used term "reflection" instead of "symmetry". Sorry for that.
For me cases are equivalent when there is an isometry of a cube that transforms one case into another.
For you cases are equivalent when there is an orientation preserving (or rotational, which is the same) isometry of a cube that transforms one case into another.

Thanks. Now I see why I get 20 and you get 22.
You treat two cases as equivalent if there exists rotation that transforms one case into another.
And I allow symmetries as well.
For example (1,6) and (1,8) are equivalent for me but not equivalent for you.

There are 4 edge-edge cases:
1) parallel sharing face
2) parallel not sharing face
3) sharing edge
4) not parallel and not sharing edge
I think you missed it.

The argument is not that infinity number of balls were removed, and it is not about 'infinity - infinity' tricks.
The argument is that each particular ball was removed at specific moment. You can provide a formula that takes the number of the ball and gives the exact time it is removed. Each particular ball is inserted once and then removed once before noon, thus no ball can remain inside the box at 12:00.