There's really not much point in trying to answer this question, without first pointing out that the content you have quoted at length from Langewiesche's classic text "Stick and Rudder" appears to be erroneous. In actual practice, pitch inputs generally have only a very small effect on yaw "coordination" (keeping the ball and yaw string centered). For more, see this answer to the related ASE question "Why does elevator input move the turn coordinator ball in steep turns?"
But in short--
Skidding turn-- ball toward high wingtip (and toward outside of turn), yaw string toward low wingtip (and toward inside of turn). Add "outside"/top rudder to coordinate.
Slipping turn-- ball toward low wingtip (and toward inside of turn), yaw string toward high wingtip (and toward outside of turn). This is what normally happens if we roll the airplane into a turn without using the rudder. Add "inside"/bottom rudder to coordinate.
This situation is also considered to be a "slip" or "sideslip" -- linear flight path, ball toward low wingtip, yaw string toward high wingtip. If this an intentional maneuver, pilot is undoubtedly holding "top" rudder, and "bottom" aileron.1 Adding "bottom" rudder is one way to end the slip.
This situation is arguably ambiguous as to whether it should be considered a "slipping" turn or a "skidding" turn-- rudder hard over toward high wingtip, flight path curving toward high wingtip, ball toward low wingtip (and toward outside of turn), yaw string toward high wingtip (and toward inside of turn).
- The degree of "bottom" aileron required to neutralize the net roll torque in this situation, and in any other slipping or skidding situation, is highly dependent upon the slip-roll coupling characteristics of the aircraft. In the vast majority of airplanes, some "bottom" aileron would be needed, but in a mid-wing aerobatic airplane with zero dihedral, little or no "bottom" aileron might be required.