I don't really care if the seat is up or down.
Yes, I've fallen in before, but I don't make a splash landing, I just get a bit more cold enamel than I bargained for. I just haul myself back up, do my business and go on my merry way.
Not a battle I think worth the trouble.
However, when no men are in my life, I keep the top down and lid shut because my Kenobi cat likes to play in the toilet, then leave wet puddles all over.