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.
You know what they say: you have to rush when Mother Nature calls. Under those circumstances, it can be difficult to read the instructions first. Good to know you saved the poor tourist a free chemical bath.