@bulba, well, it’s for the hygenic reasons that I mention putting the seat back up. Many times, a flush can back-spray and wet the seat. People don’t notice, walk away, leaving that mess to possibly dry on there, and the next person wouldn’t know.
Little kids, and lazy guys, often don’t lift the seat before taking a leak and bad aim, or back splash, can again wet the seat if it’s down.
I understand women don’t like falling in accidentally, but it seems to me that leaving the seat in the up position when not in use is the cleanest place for it to be.
If you want to keep pets/kids out of the water altogether, the seat alone is likely to be no deterrent at all… put the top lid down. Actually, that might be your (weakly written) point… and I would agree that always “lid down” is better than always “seat up”, given the choice.
(All this is from a home-use perspective… public restrooms demand another level of personal hygenic protection).