The fact that your sister said 'she's doing that stupid dance again', suggests that the dance is a pretty frequent occurrence. Yes kids like to dance but there is a time and place and when adults are trying to relax, have a cuppa, have a conversation is the time to take the dancing into the garden. Your sister could have phrased it more nicely but I suspect if she had said 'could you stop dancing for a little while whilst we have as it down, you might bump into us' your DD might till have felt she was being told off.
There seems to be a bit of a misunderstanding on her part as to what is a telling off. Being asked not to do something is a request which may become a telling off if the behaviour persists. Did the tellings off at martial arts consists of 'pay attention DD, don;t day dream' and ' please don;t touch the piano' (not tellings off) or were they 'why on earth can;t you pay attention, and listen you, are very rude' and 'why are you touching things you know you shouldn't, keep your hands to yourself, how many times do you have to be told' (Telling off).
I think a bit of discussion about the difference might help your DD cope a bit better.
Your Dad was unkind about the baby voice - it is annoying but he could have had a word with you, rather than humiliate her, or spoken to her privately and kindly.
I don't think the wanting to pretend to get a boat and play castaways is babyish - that seems quite normal to me, and also sounds like fun.
It seems your sister, your dad and your niece are all persistantly calling her babyish/immature needs to grow up - That to me suggests they have been talking about her and come up with this way of describing her personality. people get cast in roles in families and it sounds like your DD has been labelled the babyish one - which means that everything she does will be cast in this light, and when she does something more grown up it will be ignored or misinterpreted because the constructed narrative is that she is immature.
Like most 9 year olds, I'm sure she has times of being a bit immature and times of being quite sensible and thoughtful. Yes she probably needs to think about where and when she whirls about, but whirling about is not in itself a bad activity as long as you do it with consideration. I think it is also possible that she senses some people find her irritating and is trying a bit too hard to get their attention and to get them to like her and going about it the wrong way. Which very much suggests a lack of confidence in herself.
My neice was a bit like this as a child, in her own world, a little eccentric, found it hard to make friends, very sensitive and somewhat immature. She's now nearly 40 and is one of the sweetest kindest most thoughtful adults I know. I'd rather my child was a little bit different, a little bit socially awkward, than was someone who says nasty things to others to hurt them. If your sister, dad and neice have been discussing your DD and have all decided to keep pointing out she is immature, then they are bullying her.