You can acknowledge a mistake, apologize for it, while maintaining your dignity. You may believe that your apology won’t occur to the recipient as genuine unless you visibly display your remorse and shame.
This is not the belief of a healthy, non-codependent person.
In a codependent person, who makes people her higher power, the motivation to apologize comes from unconscious thoughts like, “need this person to like me,” “need this person to not get mad at me,” “need this interpersonal situation to go smoothly,” “need this person to approve of me before I can relax.”
The background motivation in a healthy person, who isn’t living for a person but rather living for a purpose (some larger end goal other than simply making, or keeping, others happy) has an entirely different thought process because survival is not perceived as threatened as it is in a codependent person. His thoughts after a mistake might sound something more like, “Oh interesting, I made a mistake. Let me take a look and see what happened. I wonder what I can learn here so that the purpose or end-goal is better served next time. I’ll speak about it with my team, or my partner, so they know I’ve acknowledged it and have found a solution since I know they also care about the purpose or end-goal.”