Buddfox - have you considered that, by focusing so much on your dd's hairstyle, and your disagreement with your wife about it, you may actually be making it more of an issue than it needs to be, and this will create exactly what you want to avoid, in the future? If you have an intense focus on the style you think she ought to have, and fighting about it with your wife, YOU are giving the whole matter a disproportionate importance, which works against what you actually say you want.
As other people have said, hair is one way we express our individuality, and there is nothing wrong with spending time, effort or money on fancy hairstyles (provided that you can afford the time, money, effort).
You don't want to make your daughter feel she has failed you or let you down, if she wants to get a weave, or a fancy hair do, or just grow her hair long, do you? But the clear impression you are giving here is that there is only one style YOU would consider acceptable for her - a buzz cut.
Can you see that that is just as prescriptive as you think your wife is being?
As your dd grows up, she will have her own ideas about how she wants to style her hair - she may well try out lots of styles, and you won't like or approve of all of them - but it is her hair and her choice, and she will be working through the process of finding her own style - and she will be better able to express her choices.
I would advise you to back off on this issue. As long as your daughter is happy with her hair style, and it is easy enough to care for, then that is what matters. Let your daughter know that you support her choice, whatever that choice may be - I get the definite impression that you are only supporting her choice at the moment, because she is agreeing with you, and if she were saying she loves her hair the way her mum does it, and wants to keep it that way, you'd be saying she is too young to make that sort of decision.
For what it's worth, I think everything I have said about your daughter applies equally to your son. Gone are the days of there being one style for boys and men - the 'short back and sides' haircut - there are all sorts of styles out there for everyone, and men and boys can express their individuality through their hairstyles just as much as women and girls can.
I hope you would love your children just as much, whatever their appearance - but I worry that having the very rigid belief you have expressed here could lead to your children feeling that your love is conditional on them having a hair do that YOU approve of. I am sure you don't want this.