Your daughter doesn't respond well to things that appear to her to be ultimatums.
You would like to protect your daughter from things like getting sick.
Here are some things that have helped me live with a bit less drama with my headstrong child:
a) pick your battles;
b) (only for things that are truly of vital importance) experiment with offering choices ("Do you want your blue coat or your red windbreaker?"), appealing to her expertise ("Molly, could you check the thermometer [step outside and see how cold it is] and come and tell me whether I'll be warm enough in my gray jacket, or whether I'm going to need my winter coat?", and making solutions available without facing off with her (quietly taking along something that would fit her and provide a reasonable amount of protection from the weather);
c) get really good at communicating I-messages ("I'm so looking forward to your concert on Wednesday, and I want us all to stay healthy before your big day");
d) find ways of coping with your own anxiety (e.g. remind yourself that a strong-willed girl is going to have an easier time resisting peer pressure to do self-destructive things in her teen years);
e) let go of the tug-of-war rope as much as possible;
f) create many opportunities for her to make decisions and orchestrate things;
g) focus as much as possible on the joy your daughter brings you.
I've learned these things from reading about Oppositional Defiant Disorder, and through trial and error. I'm not saying your daughter has Oppositional Defiant Disorder. I'm just saying that if you google that topic, there will be lots of articles that could be helpful for you. Here's an example: http://www.aacap.org/aacap/Families_and_Youth/Facts_for_Families/Facts_for_Families_Pages/Children_With_Oppositional_Defiant_Disorder_72.aspx
- Always build on the positives, give the child praise and positive reinforcement when he shows flexibility or cooperation.
- Take a time-out or break if you are about to make the conflict with your child worse, not better. This is good modeling for your child. Support your child if he decides to take a time-out to prevent overreacting.
- Pick your battles. Since the child with ODD has trouble avoiding power struggles, prioritize the things you want your child to do. If you give your child a time-out in his room for misbehavior, don’t add time for arguing. Say “your time will start when you go to your room.”
Set up reasonable, age appropriate limits with consequences that can be enforced consistently.
Maintain interests other than your child with ODD, so that managing your child doesn’t take all your time and energy. Try to work with and obtain support from the other adults (teachers, coaches, and spouse) dealing with your child.
Manage your own stress with healthy life choices such as exercise and relaxation. Use respite care and other breaks as needed.
Edit after question expanded:
There's a wonderful book about child development, The Magic Years by Selma Fraiberg, that I want to quote from (starting on p. 57):
I once watched an eight-month-old girl for three weeks as she subdued an obstinate tea-cart. She could climb on to the lower shelf of the cart, but the cart perversely moved when she did. After days of futile trials, she finally learned to tackle the tea cart from the back which rested on wooden gliders instead of from the front which rested on wooden gliders instread of from the front which rested on wheels. Now she was on it. But how to get out? It was too large a drop to climb out of the cart and her pride was hurt if she was helped out of the cart. She usually fell on her face through any of her own methods of debarkation. But several times a day she set out for the cart, solemn and determined. As she started to climb on to the lower shelf she whimpered very softly, already anticipating, we felt, the danger of getting out and the inevitable fall on her face. Her parents tried to discourage her, to distract her to other activities. It was too painful for grown-ups to watch. But if anyone interfered she protested loudly. She had to do it. And finally at the end of three weeks she discovered a technique for backing out of the cart, reversing the getting-in method. When she achieved this, she crowed with delight and then for days practiced getting in and getting out unil she had mastered it expertly. From this point she moved on to more daring ascents, climbing a few steps of the staircase, then a few more, and a few more, till the staircase became a bore. She tackled chairs, any kind of chair, undismayed by those that teetered and collapsed on her....
At this point Fraiberg talks about the urge to be upright and the rejection of adult attempts to assist. Then:
The baby resists these interferences with his own investigations and creative interests. This earns him the reputation of being "negative" and permits us to speak of the second year as "a negativistic phase." [...] But in the second year the child who has gained a large degree of physical independence from his mother and who is increasingly aware of his own separate body and personality is a child who can no longer be a passive partner. He has his own rhythm, his own style, and often he seems to value his difference from his mother, his off-beat steps, as if they themselves were the signs of his individuality and uniqueness. To do just the opposite of what mother wants strikes him as being the very essence of his individuality. It's as if he establishes his independence, his separateness from his mother, by being opposite. (Many years later, in adolescence, he will do the same thing. He will declare his adolescent independence by opposing, on principle, any views upheld by his parents or members of their generation.) [...]
But let's not get the impression that this toddler spends the better part of his day being negative. The trouble with a term like "negativistic phase" is that it distorts the whole picture of development. The chief characteristic of the second year is not negativism but a powerful striving to become a person to establish permanent bonds with the world of reality. [...] It's a kind of declaration of independence, but there is no intention to unseat the government.
We can run into serious trouble in the second year if we look upon this behavior as a nursery revolution and march in with full power for quelling a major revolt. If we turn every instance of pants changing, treasure hunting, napping, puddle wading and garbage distribution into a governmental crisis we can easily bring on fierce defiance, tantrums, and all the fireworks of revolt in the nursery. But the fireworks are not necessarily part of the picture of the second year. These are not the inevitable accompaniments of negativism in the second year. A full-scale rebellion of this sort is a reaction to too much pressure or forceful methods of control from the outside. [...]
So we do not squash the new found spirit of independence, but we direct its pursuit along other lines, encouraging it where it can be useful in personality growth and exercising reasonable restraint and prohibition where it is not.
You've taken the first step -- awareness and wondering. You've noticed that your daughter has tremendous strength of character! Now, out of respect for that strength of character, you need to learn how to use velvet gloves with providing direction. Of course, if she puts herself in danger, you'll have to step in immediately. But if she risks losing her medal, take your time. Weigh the possible consequences of taking away her control of the situation.
Sometimes you'll be able to offer choices, or ask her questions to get the gears turning in her own head ("I'm concerned about the medal getting lost at school; is there some way you could mark it to make it easier to recover, if something were to happen to it?"). Sometimes you'll need to just take a step back and let her muddle through things on her own. And the medal might get lost.
But what a doozy of a strong young woman who knows her own mind your daughter is on the way of becoming.