Apologize to the person, not the audience. Public apologies are often for the crowd.

An apology aimed at onlookers is a reputation repair with the injured person standing nearby as a prop. You can tell by what it asks for: understanding, context, credit for being honest. A private apology has nothing to gain, which is exactly why it reaches the person it was meant for.

If you owe someone something, say it to them alone and say it first. Whatever anyone else needs to know can come later and separately. Keep the two apart and the apology stays theirs.