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.
I seem to recall that Mythbusters episode proving the exact opposite. They showed that fecal matter definitely gets sprayed all over the bathroom when the lid is up when flushed.
I watched that episode recently. They just proved that there are shit particles everywhere, and the expert agreed.
You may be right; it was about 20 years ago I watched it. Perhaps their conclusion was it happens just as much when the lid is down.