I understand your point of view, but don't be confused by feelings. Anything that gets your heart rate way up and builds muscle, strips fat, is by definition exercise. What you are really experiencing with these long rides is an endorphin rush, a sort of drug addicts high brought on by sustained exercise.

When we were children and rode we didn't understand we were getting exercise, we just thought we were out having fun with our friends. As adults though the linkage is obvious.

I enjoy cycling because I enjoy the exercise, but that is not the same "fun" that I have on my 1350cc sports bike as I push it through mountain corners. There is a particularly long curvy steep section of road I sometimes ride on a Sunday morning. 4km long and probably 1000 meters of altitude. It's a real buzz to power up that section at 80 km/h or so and I often pass groups of cyclists huffing and puffing their way up. I'm sure they aren't having the same amount of "fun" as I am