2x2: Face Turning
3x3: Deep-cut Face Turning
Is This correct? Are the 2x2 and 3x3 switched? What is the exact defenition of a Deep-cut puzzle?
First off, there is a lot of contention about an exact definition of what deep cut means.
I think everyone agrees that if all cutting planes intersect at a single point the puzzle is deep-cut. The contention arises when you start looking at puzzles with a lot of deep-cut properties but don't have cutting plains intersecting at a point.
But... everyone agrees that the 2x2x2 is deep-cut and the 3x3x3 is not. For the 2x2x2 all cutting planes intersect at a point, the planes all divide the puzzle into two isomorphic groups, etc.
For what it's worth, I've been using the term "grip" to generically describe any turning point on a puzzle. For hybrid puzzles that are, for example, face + vertex turning, having a generic term that describes things to turn is nice. For example, a Dino 3x3x3 has 8 vertex grips and 6 face grips. When you're analyzing pieces in really complex puzzles you generally look at the grips that will move each piece. Looking for symmetries in the grips allows you to learn a lot more about the piece type. This also allows you to say that the corners of a Master Pentultimate behave like "deep-cut" pieces even though the overall puzzle is not deep-cut.