An apology rehearsed for an audience can miss the person it was meant to reach.

An apology aimed at onlookers is a different act from an apology aimed at a person. Once an audience exists, the words start doing the work of restoring your standing, and the injured party becomes a prop in a scene about your character. They can usually feel this, which is why a public apology sometimes leaves someone angrier than silence would have. Being forgiven and being seen to deserve forgiveness are not the same errand.

Say it to the person alone first, without the framing you would use in front of others. If a public correction is also needed, make it plain and separate.