While I agree with your first point that it is important to first admit a mistake before thanking the person for helping you with new information, I would challenge the belief that because crying in front of another person is often seen as improper in certain social situations that it is something to be sorry about. Perhaps the idea that crying in front of others stems from an unhealthy society and the belief that is if improper to do so is actually harmful to people who feel they must repress themselves in this way. By not apologising, even though there is often a strong conditioned impulse to do so , we have to opportunity to stop reproducing these kind of beliefs and help to create a world in which everybody feels safe enough to not only express there own pain but to sit with the pain of others.
While I agree with your first point that it is important to first admit a mistake before thanking the person for helping you with new information, I would challenge the belief that because crying in front of another person is often seen as improper in certain social situations that it is something to be sorry about. Perhaps the idea that crying in front of others stems from an unhealthy society and the belief that is if improper to do so is actually harmful to people who feel they must repress themselves in this way. By not apologising, even though there is often a strong conditioned impulse to do so , we have to opportunity to stop reproducing these kind of beliefs and help to create a world in which everybody feels safe enough to not only express there own pain but to sit with the pain of others.