"Old" people saying looks don't matter are not being honest
This is 100% true.
OP, please stop bothering your daughter about the attention she is paying to her appearance. Either it is important or it is not. If it's not, then let her be.
She likes the way she looks when her hair is done and shes wearing makeup. That's all that matters. The world at large, including you, does not have permission to comment on the appearance of girls and women. Tell her that and mean it.
Stop all this, "Are you a sheep/ lemming" stuff. She doesnt have to be you, and you need to see the irony in asking her to resist pressure while piling it on yourself. You're going to drive a completely unnecessary wedge between the two of you if you keep this up.
When my oldest DD was about 12 (20 years ago) my Dsis gave her a book by Bobbi Brown called Teenage Beauty, looking at makeup and choices around appearance. It gave her permission to grow up and explore aspects of herself that she had given little or no thought to as a child.
I think you need to give your DD permission to explore like that. Create a sympathetic atmosphere at home, a safe space in your relationship where your daughter can be herself and look whatever way she wants to.
Don't try to tell her who she is or who she should be. The teen years are hers to figure this out for herself. Be a friendly supporter of her steps out of childhood. She will take these steps either with or without you; you can't hold back the tide.
It will be far better for your relationship with her in the long run if you become a friendly supporter, someone who 'gets' her concerns, not Soapbox Mum. Don't take closeness to your daughter for granted.