M.P.
Do I understand what you said? You asked her if she wanted to potty train and she said half of her wanted to and half of her didn't want to. Is that right? And how early was she born? Premature babies usually reach milestones at dates based on the full term birth date. And, of course, all babies/children develop at their own pace.
If so I think you've put too much pressure on her to "perform" as half of her said she wanted to do. It's possible the half that said no is the part that's working. The whole situation is too intellectual put this way.
I'd back off officially potty training and literally leave it up to her. This means not giving her an intellectual choice which puts her in an area unrelated to her physical readiness to potty train. She's made a conscious intellectual decision. When she says half and half she's indicating there's an emotional component too.
I wonder if, as a result, potty training has become a sort of power control issue.
I'd tell her she can use the potty chair or the toilet whenever she wants to use it and then let it go for awhile. Perhaps a couple of weeks at a minium. Plan for potty training to be a no big deal, whenever you want to do it sort of feeling. Don't mention potty training at all.
Then talk up potty training as a big girl thing to do. Don't ask her what she wants. Just let her know that big girls do this.
Sometimes, not actually doing potty training is the easiest way to potty train. I have a friend whose 3 yo daughter was still wearing diapers when she wanted to put her in pre-school. However, all of the convenient pre-schools required that the pre-schooler be potty trained.
This mother had a "natural" approach to child raising. She put a potty chair in the bathroom along with several toys. Everytime she (the mother) went to the bathroom she took her daughter with her. The daughter sat on the chair, still clothed, and played with her toys. They left the bathroom door open so that the daughter could go in and out to sit and play any time she wanted to do so.
In a couple of days her mother suggested that she take her panties down to see if she could pee like her mother did. The daughter was using the potty chair all of the time within 2 weeks and able to wear big girl panties at pre-school.
An important part of potty training no matter what method you use is to not put pressure on the child. If she doesn't pee it's OK and to show that it's OK in a way that the child feels that it is no big deal. It really is OK.
But then to praise the child when she does pee. Some kids need a "big potty dance." Other's need a simple but enthusiastic, good job sort of compliment.
I do wonder if there could be a medical component involved. If she's truly drinking enough fluids I think that she shouldn't be able to hold it that long. Even over night until 10:30 or 11:00. To hold urine this long does set up, even an adult, for the possibility of UTIs.
I suggest checking with her pediatrician about both her ability to retain urine and her readiness to potty train.