We try, for ourselves and with our children, to make a distinction between "I'm sorry" and "please forgive me" -- saying "I'm sorry" is an expression that you wish that something hadn't happened, or had happened differently. It's not (or at least it shouldn't be) an admission of a moral failing.

Asking for forgiveness, on the other hand, is admitting that what I did was wrong. That's not necessarily the same thing that I'm sorry about -- as an example, "I'm sorry that ball hit you on the head, please would you forgive me for not being careful enough?". This separates my actions from their consequences in a way that acknowledges their affect on other people.

And I'm quite glad that in most situations I find myself, someone being willing to admit when they are wrong is considered a positive. It builds trust, and shouldn't result in losing face.