Why is it easier to balance a non-moving bike standing up than sitting down? When I arrive at a traffic light that is red I usually just stop the bike, stand up and balance it for a minute or two before going on, without ever putting a foot on the ground. Learning to balance a bike like that was easy, I estimate it took less than hour before I could stand indefinitely (well, the static load is quite high so I probably won't stand more than 5-10 minutes due to exhaustion). With some effort and luck I can also release one hand.

BUT, now I have tried to learn to balance the bike sitting down on the saddle and that is basically impossible. Why is that so much harder? Any hints on how to do it?
Just speculation, but I suspect that when we balance standing up, we move our hips laterally to make fine adjustments to the relative position of our center of mass over the wheels. When doing so, we push laterally from our feet on the pedals, which are low enough and far enough below the center of mass to make adjustments relatively easy to accomplish.

With your backside planted on the seat, the main mechanism to laterally adjust your center of mass is to move your shoulders and chest relative to your hips. Now you have a smaller mass on a relatively short lever anchored relatively high up. So a movement of the shoulders mostly results in a countermovement of the hips in the opposite direction. That makes it not only fairly difficult to get much movement of the center of mass relative to the wheel-ground contact, but also difficult to do so in a controlled manner.