Parents get stressed and unreasonable sometimes. If you ever want a good relationship with anybody then you're going to have to learn to deal with that sort of thing once in a while. What you should have done was - once you realized that you missed a message - seeing that your mom was irritable about it, just apologize for not noticing the message. Then none of this would have been an issue.
Besides, it's not like you are acting any better. By your own account you yelled at your mother, slammed doors, refused to just write down a schedule, then went online to call your mother names and rant about her. I'm not saying that your mom did nothing wrong, sounds like she has some anger issues to take care of, but that was out of line imo