The problem happens if "Cycle off" is set to "Overlap / Merge selected regions".
I tried setting it to "Merge regions" and that seems to get rid of the problem!
It says NOTHING about this when you read Apple's information about this.
It comes at a price though: Now regions won't be automatically merged together, if you record on one track, having already selected a region on that track. You have to do that manually afterwards. Which is why, I guess, I enabled this function to begin with. Also, it must be a bug. I doesn't make sense. I wish someone in the loop would report this to the Logic team.