For the 1st one, your description is starting to delve into the answer for the question"Why do you want to go to [SCHOOL NAME]?" I think school mission/history would fit into that question not "Why should we accept you?"
So do you think I should write about my motivation for becoming a doctor for the 1st one? I talked about the extensively in my PS so I don't really what I can add to it.
I agree with your description about the 2nd one, which is how I wrote my answer.