FWIW, I had the same problem until I realized one thing - I like laying down on my stomach with laptop in front of me, knees firmly pressing into the sofa. As it seems now this caused prolonged dislocation of kneecaps which caused problems later when biking or doing similar cardio. Two weeks after I stopped doing that the bike problem went away.
Any high rep stuff to muscle fatigue I've tried causes pain for several days. But strength training helps it in general and makes it a non-issue in day to day life.
As always, YMMV.
A year ago I got into a gym and I feared that my knee would break down, amazingly I learnt what you refer to, that weight lifting is one of the best ways to strengthen muscles that are important to the joints. When I thought of weight lifting I just thought of stressing the body, my idea now is completely different, weight lifting is a way of activating some of your more obscure muscles. Never again I got any complain from that knee.
After about a year of physical therapy, I've definitely gone a long ways towards alleviating the pain.
(If you're riding a fixed-gear bike with no brakes and slow down by backpedaling / skidding, that's another issue entirely. Just get a front brake.)