For reasons that Linkat mentioned, I have both the seat and the lid down before flushing. Flushing creates aerosols. Bacterial organisms in the toilet bowl become part of those aerosols that land here there and everywhere in the bathroom. That's why my ideal house would have a separate little room for the toilet. Like houses of old in England. And likely other European countries too.
One other little tid bit to mull over....if you ever dry your hands with those air dryers in public restrooms, just think of all the bacteria that are landing on your hands!
The air intake for those things includes the aforementioned aerosols.