With this crank, we are still talking about BSA threaded bottom brackets. "Two piece" refers to a crank where the drive side and axle are one piece,and the other piece is the non drive crank arm. The axle is usually larger diameter and hollow, and the non drive arm attaches by a pinch bolt. The pinch bolt crosses the slot and compresses the slot when it is tightened. Pinch bolt is different from the classic tapered pin (cotter pin) on a 1950's bikes such as Glider. On the cotter pin design a tapered bolt jammed against a flattened part of the spindle.
 - the arm is fitted over the spindle

What does he mean by "compression slotted"? "Compression slotted" refers to slots that allow the crank fitting to compress when you tighten the pinch bolt.

Instructions:
 He has a cut-away that shows exactly how the puller works. It is really important to tighten the crank really tight.

Relevance: One of the bikes in our fleet has a two piece crank, so they are relatively common.

Terminology:
  Two Piece
  Pinch Bolt
  Compression Slot
  Pin spanner (two pins, to remove some dust caps)