Correcting someone in public wins you the moment. Helping them in private wins you the person.

A public correction is aimed at the audience as much as at the person, which is why it lands as humiliation regardless of your tone. You get a room that agrees with you and someone who will now defend their position instead of changing it. The private version offers none of the reward and nearly all of the effect.

Hold the correction until you can say it one to one, even when the error is sitting there in front of everybody. Fix the immediate problem without naming who caused it. Have the real conversation later and quietly.