I don't know how you tell you this because I really don't want to offend you, but straight forward, your description sounds overthought in my eyes.
She is just four and she had two poopy accidents at daycare in six months? Other than that, she has a urine accident every second week there.
At home, she has no set schedule and has accidents.
I would get a timer, get some advice from the daycare how long she can hold it and have it set at home to this time. That way, you both know its time to go when it goes off and you can set it right away for the next period. In addition, I would set up a chart in the toilet where she gets a star for every time she goes and once she has a week full of stars (maybe allow one accident in the first week), then give her a big reward that she really wants.
I would try this for at least four weeks. Once she has success with no accidents during the day, you can start talking about nighttime. Try to find out when she goes at night, my son always had a warm diaper in the morning which meant he had just gone in it before he got up. That showed me he could hold it thru the night.
The part about only having urine leaks and not full accidents I cannot give any advice on, that never happened with my boy. I would ask the pediatrician about it, it could be maybe an UTI?