More years ago than I care to remember , I flew an M20C out of Andrau in Houston. (Does that still exist - it must be sky-scraper City by now?)
It had a very similar arrangement to the one that you describe BEX. IIRC The red bit is the safety gate.
You move that out of the way, and then p.. pu...pull the lever toward the tail, as it gets to about 45 degreesfrom vertical, you roll your wrist around and push it down to horizontal - this manually retracts the gear. Scooter boy - you are so fortunate as this is the only system on the C model! And it is bloody difficult. There is a low gear op speed but as you get near it, the difficult procedure is made more difficult by drag.
I did about 50 take offs and landings in that 20C and although I loved the x-country performance, I always worried about getting the gear up AND getting self and pax through the single door in a hurry, if I failed to get it down!!
Hope that helps
Stik