Home toilets use the bowl filling up to create a siphon through the drain trap. For that to work your toilet must first raise the water level above the top of the trap to create the pressure with its weight to start the siphon. I.e with the lid closed you are forcing more air to compress in a smaller volume generating a pressure difference outside the bowl and internally. Leave the lid open and that volume of air goes directly into the room without much resistance creating less pressure in the bowl compared to the surrounding air in the room. Hence with the lid open the distance, the spray travels is lower as it has a lower velocity.
That air pressure doesn’t mean anything without aerosolized particles. High rate commercial toilets create those particles and spew them out at like 6 ft/sec. Draining the tank into the bowl does not create much except maybe a few larger droplets once the drain takes most of the contents that can’t go as far, and that is mitigated by closing the lid.
Correct which is why in addition studies show that having the lid down increases the concentration of those aerosolized bacteria and increases the distanced traveled while also allowing them to linger in the air for up to 11minutes longer. The lid down causes the particles to break up into nano particles which are not visible, but linger longer and spread farther.
It appears they studied a shared university toilet, likely not a common household unit. The further study that it references I found here
This is referencing hospitals, again not home toilets. I’ve yet to see any actual info on standard low pressure toilets.
