Considering that when the toilet is flushed, many particles of whatever is in the toilet become airborne as they are splashed out of the bowl, it is more sanitary to put both the seat and the lid down prior to flushing.
However, the best answer is to realize that other people are not going to always do it your way and you should forebear and forgive one another.
To men: women are silly and unreasonable over this. Be courteous and put the seat down.
To women: Realize men are pigs and won’t put the seat down. Stop whining.
That's the way I see it.