I don't want to upset her but I have several concerns. People typically do not try to kill themselves over a talk. This is not enough. Which leads me to believe that either she has an issue that leads her to be very oversensitive or she cut herself as a way to get attention. If she wanted to really harm herself it would have gone down a whole different way. People also do not cut themselves short of actually doing enough damage to be in the hospital. Either way it seems that she wanted your attention. That is okay - she needs to find other ways to do this. See if she will go to a counselor. If the attempt was genuine then she needs to see someone, and if it was for attention, she needs to see someone. It doesn't make her a bad person.
As far as the ex, I think that this is a double standard and not fair. Lots of girls do not want their guy seeing his ex, and I kind of agree. But you can't then go and do the same thing. You both need to sit down as a couple and resolve these problems before they are so huge that she does something drastic
I am saying that if she wanted to harm herself, the attempt would have been more severe, she probably wouldn't have gone for medical attention or make this known to others. This is a sign of an attempt but not a real motivation to kill herself. These same people can be very depressed and cut themselves but not to the point of dying. It is great that you are so supportive but you really need to encourage any and all mental health treatment because this is significant and a sign of chronic depression.
I understand your ex isn't about control. many women would be uncomfortable with this and you did more than compromise. It can be a double standard but she may be more insecure then you so it isn't the same in the end. She may be jealous but decision is yours. Is it important enough to bring this up again. If it is that's fine
She probably was exaggerating but don't take that chance. Get her in treatment. It's not dangerous to show you care. She cut herself ; that person needs counseling. Don't be scared by proactive. You are NOT to blame for her behavior. That is a sign of someone that needs counseling. You should not feel guilty. You can't dictate anyones behavior. I hope this helped.