You have it mostly right. The batch in bay three would rest and mature for the duration of time you’ve set for each cycle. Turning isn’t strictly necessary for the process to occur, though it does have its benefits, and the action of moving the piles from one bay to another is likely enough to aerate and homogenize the materials.

The resting period for the oldest batch allows for less thermophilic biology to begin colonizing the compost, and these will help you continue the breakdown and release of the nutrition available there.

You might be interested to read up on Su-Johnson Bioreactors for another composting method that doesn’t emphasize turning the same way the Berkeley method does.