My son has Spastic Diaplegic CP. He too had a traumatic birth, his CP is a direct result of this. He was starved of oxygen in the 20hrs prior to birth and then was not breathing when he was born. This was a factor for him when they diagnosed his CP.
CP is when the signals from the brain to the muscles in the body are not travelling properly for whatever reason and cause physical problems, such as high or low musce tone. (there are more manafestations of CP, balance and co ordination being some) In my sons case he has a cavity in his brain and a lesion which are confusing the signals to the muscles causing high muscle tone in various parts of his body. His balance and co ordination is also terrible. When my son was diagnosed I was told that for a CP diagnosis you needed to see some form of brain damage on an MRI scan or ECG. But have since learnt that this is not the case for everyone. All stories are different. A neurologist is a good place to start for a diaognosis though.
CP can be very severe or very mild. It can mean you can't walk / move / talk or it can simply mean that you find using your fingers for fine motor skills difficult. CP is the umbrella term, it seems to me that everybodys exprience of it is different.
The only thing which might help you might be Physiotherapy. Can you access this without a diagnosis? If you want to persue a diagnosis then it might help people be more understanding, especially as she is high functioning. BUT its not important if you know her issues and you have all the help she needs in place.
There is alot of information about CP online, have a look and see if she fifts. If you feel you need to persue it then I would see her doctor and express your concerns. Its not too late.
Feel free to ask me anything about our journey with our son. I'm sure others on here will help too - they've been great with us.
Hope this helps.