Hello,
I don’t understand how the “intersect boolean” operation works (or maybe this is a bug?). I have a cylinder and a plane (scaled a bit):
I think you’re right. It seems that even when the separated part of the object is non-manifold, the intersection still glitches.
Exact doesn’t really work, I always use Fast
If you think about Face’s direction alone, wouldn’t the EXACT be the correct result. 
Since it’s a single face, I think the back can be ignored. 
Unfortunately, I’m having difficulty understanding what you are trying to say.
Basically the result you got from exact is what you would usually want because the result you got from fast is non-manifold. Non-manifold geometry is harder to work with and is not really usable. But basically boolean works better when both the object and cutter has thickness.
Ok, it has meaning for the very first case - when I’m doing an intersection with a non-manifold face without thickness. But what about my second comment, when the intersection occurred between a cylinder and a plate with thickness?
Well… it’s simply how it works… 
I would need to see what it looked like before you used the boolean. Boolean needs a clear inside and outside of the mesh.It won’t function correctly when the normals aren’t facing the correct way or polygons intersect in a way where Blender can’t tell the inside from the outside.
Unfortunately this forum doesn’t allow me attach files. But as statistics says at this image:
The plate has 8 vertices and it is just a simple cube scaled by Z.
It’s very simple to simulate this: create 2 scaled cubes and delete a face from one of these.
I’m not sure what you’re looking for at this point. One Boolean method works for you, one doesn’t, so use the one that does. Talking about how they work isn’t going to change anything, they are what they are
It’s hard to explain properly (language problem), but you can do a simple test. 
The three objects are in the same configuration. The bottom of the box is open and the plane is in the middle of the box.
※ The blue color of the face shows a specific direction toward the outside and the red color toward the inside of the face. (This is the part described above.)
fsf.blend (1.1 MB)
Test this configuration with a slightly different setting.
The box has no thickness, and the plane has thickness.
In this case, the two objects combine perfectly.
Both the plane and the box have no thickness.
In this case, the bottom of the plane is cut.
The box has no thickness, and the plane has thickness.
In this case, the bottom of the box is cut and there is a closed plane.
Also, in this case, the face orientation of the plane is reversed.
For Experiment 2 above, changing the face direction of the plane will delete the top.
For this reason, it can be seen that it is cut according to the face direction.
The answer is simple - as a novice, I constantly stumble upon many different problems.
I don’t want to solve just this particular difficulty, but try to understand basic principles - this is much more useful and interesting.
Thank you, although I don’t understand how you made these examples right now, I will try to figure it out.
