I don't think there is any one set way to apologize and it depends on the person you're apologizing to. Some people require more of an effort than others.
You just end up saying sorry and explaining what's needed and making that person feel better and or whatever the outcome is. It depends on what you did wrong.
For me, I don't need any special process to accept an apology. Just saying sorry is enough apology to me. I don't have to make it difficult for people to mend fences.