This seems like something that should be true, but I think I remember seeing a Mythbusters episode where they decided it didn't make a difference. That show was more about entertainment than science, so I wondered if there was a more rigorous study done? I've definitely seen splashes of water(?) come out from flushes so that alone seems to argue for closing lids.
Unfortunately, I've never been to France, but nothing beats a spotless clean public toilet. And, thanks for the tip: if someone holds the door for you, kindly step back. That alone should be highlighted in all tourist guides.