I think what you wrote was good and at this point it would be best to give her some cooling off time. Pressuring her may backfire. She is hurt right now and is questioning the relationship she had with you based on your response that day. Rather than reply harshly it would have been better to explain why you did not owe any money. Regardless, the mom did not handle the situation well because rather than contacting you to discuss the situation she rudely texted you and demanded you pay her back. Therefore, she has fault in this even though you also did not handle your part well. Regardless of what happened blaming or pointing fingers will not help at this point. Everything you wrote was wonderful and I think you did all that you can do. It seems that you are very sorry and want to get passed this. Like I said give the mom some time to allow this to pass and then try contacting her again to set up a time to meet and discuss this. After some time you can also ask your girlfriend to try to arrange something and have her tell her mom personally how you are feeling. Try to be patient and not worry about this and that it ruined everything. This can be fixed. It just may take a little time, but it will be well worth it.
I wish you the very best and please let me know if I can be of further help.