Welcome, Hope Rising.
Love your name!
I'm sorry about the pain you're having to process in your therapy. It was brave of you to talk about something that you feel so much shame about, that you don't want it written down (I can relate to that). To me, your asking her not to write that down was entirely reasonable, especially considering how much pain you were obviously in at the moment, and also considering how much shame you were feeling. I'm not sure what disturbs me more about her response to you - what she said, or the way she said it.
I really don't understand why, in the context you described, she would take your request (which she should have responded to the first time, and not made you repeat yourself) as an attempt to "control" her, or why she would speak with such coldness.
I have no idea what to make of the "side hug" either, but it surprises me, too, that someone who describes herself as "huggy" would hug that way!! I can totally understand why you would have been disappointed.
The T I most recently worked with would most definitely have respected my wishes not to write something down, if I asked her not to. Actually, she hardly ever took notes, and never, ever when I was in the middle of expressing the kind of distress you described. She would have been paying attention to me instead. As for hugs, we didn't hug every session, and she struck me as not the huggiest person, but her hugs felt warm and caring to me.
I do not think I could work with a T who treated me the way you described yours as treating you. Given the first two situations you described, I would also not feel safe enough to risk pushing her away. I don't know how long you've been working with her, or if you've ever worked with another T. But I'm hoping that the feedback and support you get here can help you decide what to do next.
Peace,
SG