The person who corrects you privately has spared you work that a public correction would create.

A private correction gives you one job: fix the thing. A public one gives you three, because you are also managing what the room thinks and what you look like while you take it, and the fix becomes the smallest part of the afternoon. The cost of receiving it is not the same, and whoever chose the quieter route absorbed some awkwardness so you would not have to.

When you spot someone’s mistake, take it to them directly rather than into the group thread. Keep it short and free of an audience. And if someone did that for you lately, say thank you, since nobody notices the day they prevented.