I wouldn't get her contact lenses, if two opticians have said the same thing. She would really regret it if wearing contacts at 14 jeopardised her chances of wearing them when she is older.
I have eyesight as bad as your daughter (I think it was about -7.50 when I was 14, but is now -.8.50 in one eye and -8.00 in the other). I really do sympathise with her. It must feel so unfair. It's awful when you are very badly short-sighted, because even if you go for the ultra thin frames, they still make your eyes look much, much smaller. I used to hate the way my glasses made me look. I still feel happier wearing contact lenses.
Can I ask how on earth she manages to do anything without her glasses? I don't think I could get more than a few yards from my front door without wearing my glasses or contact lenses.
Now I think you're going to have to help her feel more confident in her glasses and convince her that she looks good in them. Has something happened at school to make her suddenly self-conscious about them?
I think you also need to emphasise that it's likely she can have contacts in the future, but if she won't wear her glasses or insists on contact lenses now, in the long run that may mean she'll be limited to glasses indefinitely.
Does your daughter wear make-up? I was beginning to use eye make-up at around her age. Maybe you can bribe her with some beauty treatments to encourage her to wear her glasses, which 'open up' the eyes and counteract the fact that the glasses make them look smaller, such as an eyelash tint, a make-up class (maybe somebody can show her how to subtly apply eyeliner to enlarge her eyes beneath her glasses) and maybe eyebrow threading, as shaped eyebrows can really bring out the eyes (although go with somebody who will make sure they look nice and natural). I know the idea of a 14 year old having beauty treatments may not be all that appealing, but if it helps her feel more confident about wearing her glasses, then it may be worth it?
If you can afford it, it might also be worth looking into getting her an obviously designer pair of glasses with ultra thin lenses as well - she might be less self-conscious about some overtly 'cool' glasses.