Sorry, but it seems that your feelings towards your daughter are rather selfish.
You want her to be "sweet and caring" and to have physical contact with you. She doesn't. Well, it's just the phase most people go through. Kids like to be kissed and hugged by their parents; teenagers find this embarassing, especially if you try to do this in public. Please be patient: in a few years, when she no longer feels the need to prove her independence, your daughter will stop running away from your hugs. Though don't expect her to be as eager as in childhood.
You are discouraging her from moving to a different province or state because you want her to be with you; you are afraid of dying alone. Well, nothing is as toxic to family relations as parents trying to shape their kids' life in their own interests. If you try to push her, more likely she'll either go full rebel and run away, or she'll yield to your arguments but later will blame you for her broken dreams.
Support your daughter. Warn her against truly dangerous actions, but otherwise let her take risks. This is your best chance of having great relationships when she grows up a bit.
UPD: I wrote this before you edited the post with the description of your fight. Well, that sure makes things more complicated, but not hopeless. First of all, you have to give an answer to yourself: why did you behave the way you did? Why did you take your brother's side instead of your daughter's? I'm not a psychic but probably all your life you were taught to respect tradition over personal rights. If you can explain your reasons first to yourself and then to your daughter, and make an effort to support her instead of forcing her to adhere to your notions of what's proper, then I believe you have a chance of repairing your relations.