Same sequence you've already outlined:
Rinse/Spin Cycle
Low Spin
Softer Signal
Warm/Cold
Extra Rinse 4 presses
It starts immediately if the door is already closed, no need to press Start.
The tech sheet stashed in the machine says Kenmore Elite HE units (circa 2006) also have the cleaning cycle but a slightly different key sequence:
Rinse/Spin Cycle
Low Spin
Warm/Cold
2nd Rinse
Cycle Signal 4 presses
I imagine whether a given machine has the cleaning cycle depends on the control board software revision.
Regards to the Bulky cycle water level, I have a service bulletin dated Sept 2005 that addresses possible leaking on Bulky (when running a small load) for stated models GHW9250M, GHW9400P, and GHW9460P ... and serial numbers </u>prior</u> to CSS05 (5th week of 2005) ... those are the machines subject to the issue at that time. The fix is to change the control board, which has updated software, version number on the label ending with 2 instead of 1. Presumably serials after CSS05 already have the update.
The version number on my board ends with 4, which corresponds with the engineering revision on the model number GHW9400PW4, although that doesn't necessarily mean there's always a match on that point.