Here are your choices:

1. Doing the same you did, produces a closed path. I have no idea how you achieved yours.
As the cutting path doesn't have any width, the boolean action merges the parts together as described in another thread.
2. Still making a same cut, this time, with a closed path on top. It merges things alltogether, at least one part is deleted.
Cutting paths work best with paths that have a fill only and no outlines.
On the image you posted, there are still problems with the "right cut" part: the path's ends doesn't have one solid edge.
That is because the boolean wasn't applied to the outline of the curve.
3. Solution1: masking the wanted part with an object on top.
4. Solution2: clipping the wanted part with an object on top.
Notice that when masking, the bounding box will be covering the union of the mask object and the masked object, while with clipping it will cover exactly the clipping object.
5. Solution3: conert outline to path on both objects, then apply boolean.
To show the methods I didn't used one "line" on top, because the result is better shown this way.
At solution1 and 2 you would have to do the same thing with the left and the right parts, and, they would be aligned next to eachother -if that's what you were after.
At solution 3 I used similar object to the masking, because I don't see the point of making a gap between the two cutted parts, but it is possible to make.