Opening the book…
A correction heard by an audience stops being about the work and becomes about standing. The person spends their attention managing how they look instead of absorbing what you said, and everyone watching learns that mistakes here are punished with exposure. Private correction keeps the information intact and keeps the room safe to make errors in.
Walk them out of earshot, or wait for the next natural gap and say it in three sentences. Correct the standard in public if you must — "we file these by Thursday" — but never attach a name to it. If you slip and criticise someone in front of others, go back within the hour and say so plainly, in front of the same people.
When something must stop this second for safety or in front of a client, stop it out loud and repair it privately afterwards. Group debriefs where everyone names their own errors are a different mechanism, and only work when you go first.