Permanent Press, the original version anyway, was a chemical process. Fabric was treated with methanol to hold its shape and resist wrinkling. Washing in warm or hot water relaxed the fabric so it shed wrinkles that developed when the clothing was worn. If subjected to the forces of spin while hot or warm, wrinkles would tend to be pressed back into the fabric. Cooling the fabric by floating it loosely in cold water counteracted the relaxation of a warm wash to minimize setting wrinkles during spin. Spin speeds typically were slow, also to avoid compressing fabrics and causing wrinkles. Of course, tumble-drying temperatures also relaxed the fabric, followed by an extended cool-down, so a wash cool-down wasn't particularly important UNLESS the clothes were line-dried.
Solid-basket and spin-drain machines were perhaps less successful at the washing-side cool down because they had to spin to throw out the warm or hot wash water. Machines with perfed baskets that spin-drain generally used a partial spin-drain whereby the basket didn't get up to full speed, then refill with cold. Solid-basket machines (GE also did this with their perfed spin-drain design) typically triggered an extended cold spray immediately at the start of spin, perhaps lasting through the entire post-wash spin period.
Whirlpool and Kenmore used different sequences for their infamous Perm Press or Wash-n-Wear cycle. Several partial drains (until the water level switch reset) and refills with cold, with no agitation. Partial drain, then agitation *while* refilling with cold, then a repeat immediately when the water level satisfied on the fill. Partial drain, pause for the timer increment to advance, then refill, then agitate for an increment, then repeat. The separate "Knit" cycle when introduced had a one-stage cooldown, whereas Perm Press had a double cooldown. As conservation of water became a concern, Perm Press changed to a single refill and the separate Knit cycle disappeared.
Consumers nowadays have largely moved away from line-drying, but perm press cycles persist on washers, I imagine mostly for marketing purposes.
