Let me put it bluntly: Teachers should apologize when they make a mistake or are wrong.
Why? Because this is the ultimate model of owning your choices. Our students look to us each and every day to show them how to navigate the world. It is often baffling to me how many third grade students still come into the year thinking teachers are perfect. I absolutely want my students to see that I am not perfect. Not by a long shot. I have very candid discussions during morning meeting about this with my students; there are plenty of people who are better at things than I am. I am also better than some people at certain things. But the fact of the matter is, I am not perfect and no one else is either.
When we own our own actions, we are teaching our students responsibility. When we apologize for saying something that was misinterpreted (or when we let our emotions get the best of us and snap at someone when we really didn't mean to), we are modeling for our students how to relate to people and get along with others even if we don't always like that person or agree with them.
I replied back to my friend and pointed out that since he owned what he did and apologized to his students for his error, the lesson wasn't the failure he thought it was. It was a success because the learning the students received may not have been what he originally planned to teach, but I guarantee they learned that making mistakes is normal and is how you grow as a person.