She is in the swimming pool
I am not completely sure.
She is in the swimming pool means that she is inside the swimming pool and she is having a bath.
She is at the swimming pool means that it is possible that she is anywhere of the swimming pool (inside, outside, beside and so on).
Yes, you pretty much have it.
She is in the swimming pool means that she is in the pool swimming, not having a bath (well, she could be having a bath, but that would not be normal).
She is at the swimming pool means that it is possible that she is anywere around the pool, in the pool, beside the pool, etc.
In England we wouldn´t use "bath" when in a swimming pool but you can use "bathe" which would mean to immerse oneself in water in your example.That is of course presuming the pool is filled, ie it might be empty, hehe.
Nila, to elaborate on Jacks answer, here in the EEUU we don't usually use the term bath/bathe to refer to swimming. Even though it is proper use we would just go in for a swim or a dip. Bathing is used mostly for cleanliness reasons or maybe sunbathing. Now I've probably truly confused you.
Ok, you have answered the question. But now the doubt is if "bathe" can be used when you are inside the swimming pool. From Eddy's point of view it is correct, but from Yesero's point of view is not correct.
How do you say that?