what is the difference between Direct insert and Drop and Catch ... [?]
The difference is a bit technical. For BH we define a commutator to be ABA'B' where A is the insertion part of the commutator, and B is the interchange part of the commutator. The A part of the commutator is generally 3 moves, let's call those 3 moves PQP', and the B part of the commutator is generally 1 move. Every 8 move commutator in BH cycles 3 pieces, one of which is called the "lone" piece, and the other two are called the "interchangeable" pieces. The interchangeable pieces are the ones that are moved with the B part of the ABA'B' commutator, and the lone piece is the piece that is
not moved by the B part of the commutator.
Now, the first part of the A part of the commutator (The P in PQP') can move different types of pieces. It can move exactly one interchangeable piece and nothing else, both interchangeable pieces and nothing else, all 3 pieces, or exactly one interchangeable piece and the lone piece.
The "Drop and Catch commutators" are the ones where the P part of PQP' moves:
1) exactly one interchangeable piece and nothing else
or
2) both interchangeable pieces and nothing else
The "Direct Insert" commutators are the ones where the P part of PQP' moves:
1) exactly one interchangeable piece and the lone piece
or
2) all 3 pieces
---------
Basically, the "Drop and Catch" commutators do
not move the lone piece on the P part of PQP', and the "Direct Insert" commutators
do move the lone piece in the P part of PQP'.
Chris
I think I almost got the corners down. Im gonna memo using visual 
