You don't want your daughter to have to conform to notions of beauty - but it sounds as if you'd disapprove heartily - and vocally - if she decided she wanted weaves/conditioner/braiding/colour etc.
So it is fine for her not to conform to society's ideas of beauty AS LONG AS SHE CONFORMS TO YOUR IDEAS?? Because that's how it sounds.
I know your children are twins, but that doesn't mean they aren't individuals. Your daughter is allowed to have different ideas about how she wants to look.
You want your dd to choose her hairstyle to please you. That is as bad as her choosing her hairstyle to suit society's ideas of beauty.
She should only choose her hairstyle to please herself.
Empower her to make her own choices - and make absolutely sure you tell her that she doesn't have to please you with those choices - because I have a horrible feeling that, at the moment, she knows what you like and don't like, and wouldn't want to make a choice that you'd disapprove of.
I do think that, when a child is young, the parent who is doing most of the hair care (ie. doing their hair each day, dealing with hair washing etc) does get a big say in what the style should be - in that the child shouldn't be allowed to choose a style that is going to take ages to do every day. And the style should be sensible and practical for what they are going to be doing - ie. long hair should be tied back for PE etc, and a hairstyle shouldn't get in the way of children playing, being active etc (I hope that makes sense). I suspect that your wife does most of this sort of stuff for your children at the moment, and that does give her opinion weight - more weight than yours, I believe. But your dd's opinion should also have weight.
But as a child gets older, and can take on the responsibility for doing their own hair, they should get more say over style. Yes, there are constraints around cost, suitability for school, practicality etc - but within those, it should be the child's decision.
You say your daughter tells you she wants her hair short - can I ask you what she tells her mum about the style she wants, when you aren't around? Is she possibly telling you what you want to hear?
I do think you also need to adjust your idea about a woman's hair being a crown of thorns rather than her crowning glory, as that is a very negative image, and you don't want your daughter to grow up hating any part of her body, do you - and she will pick up that that's how you feel, and it will negatively affect her self image.
It is worth bearing in mind that being really strict about something can have the exact opposite effect to the one you want. So by making your dd have short hair, so she doesn't get the notion that her beauty depends on her hair, you might push her to rebel against that notion - and in years to come, her hair may become the main focus for her.
To give you an example:
My mum hates pink. Loathes it.
As a girl, growing up, my sister and I had no pink clothing whatsoever. My mum made a lot of our clothes herself, so wasn't constrained by what was in the shops.
As we got older, we were allowed more choice about the fabrics she bought to make our clothes - but I knew that I couldn't have chosen a pink fabric - I knew her views all too well, and knew I would get that Look from her. It was subtle, the way we were steered away from anything she disapproved of - but we knew what was and was not allowed.
She was just as opposed to the idea of 'Girls Wear Pink' as you are opposed to 'A Woman's Hair is her Crowning Glory' - and my sister and I were very firmly pushed away from all things pink.
As an adult, I realised that I liked pink, and that I resented her for suppressing my individuality the way that she had. I wear quite a lot of pink now - all different shades - and I revel in all the colour choices available to me. And I still feel that this earns me her disapproval. My sister has completely absorbed mum's anti-pink stance - and I think that's another of the reasons why mum loves her best, and always has.
If you feel your dd is being forced into styles she hates, because her mum likes them, then of course you should do something about that. But don't do it just to enforce on her your own ideas about beauty and societal norms, please.
I am living proof that doing that has a negative effect on parent-child relationships.