Hello @PierreB, from your example I can see that you don’t need a “more or less rigid” structure to form a rotary kite. Your example uses a purely tensile structure to achieve that. But the result is in fact a rotor. A rotor, because the mechanical constraints (implemented here by the central parachute-like structure that acts as a hub) force the blades on a rotary motion.
The example of the kite flying a circular path, like the Makani kites, is different, in my opinion. Here there is no mechanical linkage (I should have probably use this term instead of “more or less rigid elements”) of the kite to the center of the circular path. The circular path is generated by active control (one can also imagine a passive control).
I think the question is (that is how I started the post;-) what you consider as kite: is it the individual blade or the rotor. The wind turbine analogy is maybe helpful: blade = kite, rotor = rotary kite. But, as I wrote earlier and what also @tallakt mentioned, these classifications are only helper constructs to somehow distinguish the different concepts. You can develop different classification systems depending on what classification criteria you apply.