It’s called toilet plume.
A toilet plume is the invisible to the naked eye, cloud-like dispersal of raw, infectious sewage particles as a result of flushing a toilet. Flush particles rapidly rise out of the bowl and several feet into the air after flushing. Aerosolized droplets can remain suspended in the air for tens of minutes between users, posing an inhalation risk. These invisible particles go on to coat and spread onto surfaces like floors and counters or objects like hand towels, bathmats, or even toothbrushes.
Mythbusters.
Do whatever you want, indoor plumbing means there is shit everywhere, closing the lid has little to no effect. This is just life, get over it.
Different is between a fart of poo water and a small geyser.
One goes upwards, one goes outwards, it’s the same mass and it travels the same distance, closing the lid is a placebo, but if it makes you feel better then go for it.
Thats what I am saying.