I haven't read the whole thread but I have read all your posts. One thing really stood out to me- her behaviour isn't about you. It isn't about embarrassing you. Her behaviour is about her.
Behaviour is communication, something about specific situations isn't working and she's developed a strategy to deal with it that's unhelpful for all of you.
So- not about you. Think about what's going on for her. I promise she isn't lying awake at night thinking up ways to embarrass you!
Secondly- all children respond individually to situations whether NT or ND. So you need to find strategies that work for you and her, regardless of what label the strategies carry.
Some ideas-
talk about behaviour kindly at other times. Have Saturday breakfast together with waffles and stuff and say something like 'I got really upset and cross yesterday when you were shouting and crying at the playground. Can you explain what was going on for you so we can do it differently next time? It wasn't very nice for anyone.'
You can come up with strategies for what to do when she feels like that.
Have some destressing activities that you do often and that she can ask for when she need it- like hot chocolate or a big squeezy hug or a bounce on the trampoline. Make a collection of those things. Do them often and remind her she can choose one of those if she's getting a bit agitated.
It's about teaching her better strategies for managing frustration/disappointment/sadness etc.
There are all sorts of ways of teaching children to be more emotionally resilient- including avoiding situations that trigger problems.
I had one that couldn't decide how to spend pocket money. School trips had to take him somewhere else while the other children went to the souvenir shop. It was a big deal and it had to be managed. That's ok. Sometimes they aren't able to handle particular things yet.