So, there is 3 windings on the case and 3 windings on the stator and only two alternating negative and positive poles propelling it forward? If I understand this right.
You are on the right track. Don't know what you are referring to when you say case vs stator. There are stator windings and there are rotor windings. The stator is the stationary windings located just under the outer case of the motor, it includes all windings in this area all the way around.
All motors have a rotor winding but this winding is the type that will vary from one type of motor to the next. This is the winding is attached to the rotating shaft in some way and moves in unison with the rotation of the shaft. In a squirrel cage induction motor these windings are simply metal bars (usually copper I think) with a shorting ring attached at each end to form a 'squirrel cage'. I think it gets that name because it looks like the exercise wheel found in pet rodent enclosures.
The induction motor gets its name because the current in the rotor windings is induced by magnetic fields from the stator. There is current flowing through these bars and shorting rings causing additional magnetic fields used to create more repelling between stator magnetic fields that cause the rotor to turn. Motors with wound rotors need to have slip rings or commutators and a contact device to transfer current to the rotor.
back to what you said with 3 windings-
In a 3 phase 2 pole motor there will be three windings - one per phase in the stator making one pole, and 3 other windings - one per phase placed 180 degrees around the stator from the first 3 making the second pole.
In a single phase motor there will be the main and aux/start windings making one pole with the same located 180 degrees apart making the second pole.