I suggest that it's not likely that a 3yo can sit still throughout a movie. It's just too long for her to sit still. I suggest that a better way to handle that would've been to get up and stand with her in the back of the theater. I've seen parents do that.
How did you talk with her on the way to the car? Were you irritated and it showed? I suggest that being sympathetic works much better. "I'm sorry we couldn't stay for the movie. I know you're disappointed."
It sounds like you got into a battle of wills with her. I suggest that she needs your comfort and not punishment. Once we got home I would've put in a movie or played with her on the floor. Find a way to distract her by offering something fun to do. After all, you were expecting her to do the impossible by expecting her to not wiggle and move around. She needs discipline which includes teaching her how to regain control of her emotions.
A social worker suggested to my daughter that going to their room is a time for the child to put themselves back together. Thus they can play with toys or even watch a short movie or TV program. When you're upset do you just quietly sit down and do nothing? She has all sorts of pent up energy that she needs to get out of her system.
I sometimes ask my grandchildren to run to the end of the hall and back. Distracts them and helps them get rid of that energy.
When my granddaughter was that age, she would often not want to leave day care. She'd cry and tantrum at times. I took that time to help her calm down. We would sit in the car with her in my lap while I sympathized with her wanting to stay. I would have a few small toys that she could play with. Rarely did it take longer than 10 minutes for her to be back in control of herself. I suggest that doing this teaches her how to let go of the anger and regain self control.
Spanking does not teach self control. If it teaches anything it's that it's OK to hit. And it increases her anger. Spanking is counter productive in this sort of circumstance. Unless you routinely give her a pop to the butt and found it to work in stopping the tantrum, I would not use it.
AFter your SWH. You're frustrated and angry. She picks up on your feelings and then feels insecure causing her to act out more. She needs your calm, reassuring presence. You're asking her to be a little adult who knows how to manage her feelings. She's only 3. She's doing the best that she can do.
I suggest that you put yourself in time out. Putting her to bed early is one way to give yourself some time to calm yourself down. But if you do so in anger she'll resist because she needs you to be the strong one who is able to manage her own feelings. Save the "I'm done" for after she's asleep.
Much later; The goal of putting her in her room is to get her to calm down. It's good that she considers it a game. Go with it so that you both get out of the battle of wills. Kids always win in this sort of battle. I urge you to focus more on what she needs and less on exacting punishment because her behavior isn't what you'd like it to be.