There have been several variations of the cool down sequence amongst Whirlpool and Kenmore models over the years.
Very early "wash-n-wear" models did not agitate during the cool down, they simply did repeated partial drains and refills through a couple timer increments.
As noted, a variation on Kenmore involved a partial drain (until the water level pressure switch reset), then fill and agitate simultaneously (at low speed) for the remainder of the drain increment until the timer advanced to the next increment. If the target water level hadn't been reached (which was the usual case, except perhaps in cases of very high water pressure), then agitation stopped on the 2nd increment until the fill level was reached at which point agitation resumed for two minutes. Repeat the sequence for a two-stage cool down, then continue with drain/spin/rinse/spin.
Whirlpool's Knit cycle (which was separate from Gentle) had a one-stage cool down. I don't know Kenmore history that well, if they carried a Knit cycle with cool down.
Perm Press was cut back to a single-stage cool down later in belt-drive production when conservation became a concern. Our LDA7800 originally had a double-cool on Perm Press, a replacement timer was revised for single. The original timer had the shift to low speed agitation at 4 mins for the remainder of the cycle including rinse. IIRC the replacement did shift to low for the last 4 mins of wash, but the cool down and rinse may have been at high speed.
Far as I'm aware, direct-drive units always had a single-stage Perm Press cool down, with no separate Knits cycle. Spin-drain models drained almost completely until pressure switch reset due to the water kicking up.