I think you have figured it out. I would highly recommend painting behind the tank and not removing it. The original plumber must have done a fantastic job. Mine used wax rings, which have started to leak, so I have a big project ahead of me.
If you look down in your tank, you will see the two side screws towards the front of the tank and one screw that is centered and up towards the back of the tank. Mine is all apart and I have been cleaning and preparing it outside to be reinstalled. That center screw goes down into a threaded brass "nut". The nut is actually a pipe that has been cut in quarters on one end and flared to hold it against the top of the back of the bowl unit. When the tank is put on top of the bowl, this helps hold it in place. Not sure why they wanted this third screw. So I think you can take off just the tank by unscrewing that screw and the two sides ones to remove the tank. You will then probably need to get new rubber gaskets. If the tank is off too long, you may have trouble with gaskets drying out.
I hope this helps. I could try to send a photo, but I think you have a good idea of how it works. The part I could not remember is if there was a nut on the bottom of the center screw, which would be accessible only by removing the bowl and I don't think one is needed because the brass "nut" is threaded. Again, I would encourage you to paint behind the tank as best you can.