Giving criticism is a delicate dance between being clear and being heard. Most people have a natural reaction to criticism of throwing up defensive and protective walls against potential attacks. This is true even when we present the feedback using all the positive, constructive and nurturing techniques of good communication and leadership. We would like to believe that if we speak truth with a kind and loving heart and a spirit of edification in just the right way, then our message will be received as we intended. This is just not the case.
People do get defensive, no matter what we do to mitigate it. And yet, as leaders, there are situations where we need to give feedback and criticism to the people on our teams. It is critical that we recognize that our role is not to "give" criticism, but rather to "communicate" the criticism. That is, we need to take the leadership of not only presenting the feedback, but making sure it was heard.
One of the ways we make sure feedback in heard, is to remove as many of the obstacles to being heard that we can. One of the biggest obstacles preventing people from hearing feedback is pride. When a person is in a group of other people, they are much less able to hear criticism because of the potential hurt to their pride. They do not want to look bad in front of other people. It is bad enough that they feel bad in front of you, the leader.
I often find myself in situations where I recognize behavior that is hurting the team's ability to succeed. Part of my job as a leader is to bring that behavior to the fore, so we can improve the likelihood of the team's success. For example, when one team member gives a customer presentation, I sometimes see another team member interrupt to correct an error in the presentation. Sometimes this works, but usually it sets a tone of discord in front of the customer that is counter to the team's success. This behavior needs to be addressed.
The worst thing I could possibly do is mention this problem right in front of the customer. That would be doing more of the same. I might consider raising the issues during the group's debrief of the meeting. (You need to have these kinds of debriefs if you don't already.) If it is a pervasive problem across the group, that might be the right solution. But, if the problem is really with one or two individuals, they will not be responsive to the criticism in the group setting.
In the group setting they will throw up their walls of defensiveness. They will argue about the importance of fixing the mistaken information we gave the customer. They may go silent, disagreeing in their heads, and not being open to changing behavior. If you can meet with people individually, you remove pride in front of their peers as an obstacle to hearing and addressing the problem. Your job as a leader is to "communicate" the issue. That means making sure that the issue is heard.
Back to the customer situation: you can always agree in the meeting debrief what the right information the customer needed to hear. If the customer needs an update, you can use it as an opportunity. Use the update to call the customer. Tell them the mistake. Build in them a sense of trust that when your team makes a mistake, they are proactive about fixing it. That is a whole lot better than building in the customer a sense that you have a team in discord.