Dennis- 30 years in the "sea of estrogen"- love it! Applause, applause to you for bravely navigating such foreign, unfathomable, uncharted, and unpredictable waters!
I think "Put the seat down" may harken back somehow to the days when men were supposed to flop their cloaks down on mud puddles so that those of us of the gentler sex wouldn't dirty our pretty little shoes while out walking. ( Me, I'd rather be carried over said mud puddle- nothing better than being lifted into the arms of a big strong man, plus the added bonus of no muddy cloak to wash later!

)
"If you sprinkle when you tinkle, please be neat and wipe the seat." Wise words, regardless of gender, although cutesy bathroom plaques has always seemed a little weird to me.....(Pre- apologies to any collectors of cutesy bathroom stuff.)