I always put the cover down, that's the correct thing to do...
putting the seat up to take a piss, is also the correct thing to do..
But... if a woman complains, i'll leave that seat up all the time, after all, is she so inept she can't put the seat down before she uses the toilet??
What about in the night, if she sat down all sleepy and got her butt wet..?? i guess she'll learn to check 1st after the 2nd or 3rd time it happens..