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.
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.