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Most of the path operations work on the area inside paths, rather than on the paths themselves. Since the 2-node path has no area, the math formula which makes the intersection work doesn't give the expected result.
Also, most of the path operations can only work with 2 objects. Union can be done with more than one selected object. And sometimes intersection can (for the proper kind of paths). But I think all the rest can only work with 2 selected objects -- Unless! We have some extensions now which will perform multiple booleans in one step, for Cut Path, Division and Difference.
Anyway, in this case, I would suggest performing the intersection on only the outer rectangle. Then the single path remains where it is, and the cutter will recognize everything as paths.
I'm also not sure how a cutter would recognize a clipping path. Might be interesting to experiment, although I don't have any cutter.