I'm 63yo with bipolar2, depression, anxiety and dysthymia. I didn't know I had those illnesses until 9 years ago but I did know I was a nervous impatient, impulsive character.
So in my 30's when I had two daughters 3 years apart, my impatience was well remembered by my kids when they became teens. I vividly recall one telling us at xmas gathering about the Golden book "saggy baggy elephant" ...I'd tell them its bed time then "once upon a time there was an elephant called Saggy Baggy...the end".
This went on for two years until one said "that's not the whole book dad". So I'd include two pages randomly selected as well. Playing with kids just not my thing and I don't believe you should punish yourself for something that comes natural for some but not you. I'm sure your daughter at a certain age will benefit from your natural way of helping her where others might not be so good at that task. I think you are being hard on yourself.
There is also the tendency to place events onto our behavior when there is no solid evidence of it being to blame. Your childhood trauma might not be the cause at all yet we want to attribute such lack of qualities like playing with our children towards it as it then makes us feel there is a reason.
All of this boils down to lack of acceptance of yourself as a person and as a parent. Yet, you are a very good parent as you have taken the opportunity to write here about your problem, most don't. Many don't care. You do.
A short story. My first wife and I had our kids. At 3yo my eldest daughter opened my bedside drawer. She wasn't allowed to. Her mother closed it, she opened it again, mother closed it...this went on for over one hour! Finally our daughter gave up and never opened that drawer again. A battle of wills. The screaming and crying, my daughter concerned me that much I asked a GP what to do- "Well, I haven't ever treated a child for harm from excessive crying". So yes, it is intolerable that high pitched screaming but you have to accept that it wont harm her and she must learn that you will only respond to her when she stops! Otherwise you are controlled.
Please google the following and read the first post of each thread
Beyondblue topic worry worry worry
Beyondblue topic what you cant change
Beyondblue topic the best praise you'll ever get
Beyondblue topic accepting yourself, the frog and the scorpion
Beyondblue topic advice please my bucket is full
I hope they help.