Yes, you're right. In my friend's case the public system didn't classify her as needing publicly funded plastic surgery (due to her age for some reason!), so when she had the op done privately, it was classified as cosmetic surgery. My mistake -I used the wrong term in my original post and I'm unable to edit the original message anymore.
However, sometimes it's there's a thin line between plastic and cosmetic surgery, so I don't condemn cosmetic surgery either if it really improves someone's quality of life even if there's no obvious medical reason for it. My ex-colleague's child had a visibly deformed earlobe, and because it was completely harmless, there was no medical reason to fix it -it too was condemned to be an entirely cosmetic issue. He was really upset about his "funny ear" because it caused a lot of questions and some teasing, so in the end, after long consideration, his parents paid for cosmetic surgery to fix it because the questions made him very upset on a regular basis. Now we all know that he should have been accepted as he was, but he just wanted to be like other kids and not always answer questions about his ear and it was a real issue for him. He was six or seven years old at the time of the operation, and was really happy when he got a normal-looking ear and didn't face any more questions at his new school.