That is pretty much the meaning of a closed system.
Sure, the poynting vector is useful but it does not tell you that radiation disappears leaving a black streak between light bulbs.
Look at my examples again. They go well beyond your 1899 idea of thermodynamics and have nothing to do with the fantasy paragraph above. I wouldn't say you are duped. Just ignorant of the formalism of thermodynamics and the observable, repeatable experiments that went into shaping that formalism.
Yes, Clausius originally referred to something like that. An 1878 translation of his words are as follows
Notice that Clausius was referring to heat, and never used the word "energy". So Pale, in the second sentence of your paraphrasing, the energy is understood to be heat energy. Clausius, of course, had no inkling of the quantum aspects of radiative thermodynamics.
The more modern entropy definition which subsumes the Clausius definition, of course strictly refers to heat energy." Pale, if you just simply use the word "heat" everywhere in your definition of the second law you will be OK with your wording.
Yes, your reference to Clausius, and Kelvin is old hat, but does not clarify radiative thermodynamics. You do need to read more about the how experiments go beyond the 1899 version, and the use of mathematical models in explaining the quantum mechanical experiments. You need to understand more about how quantum mechanics has influenced the modern concepts. If you want to progress in your understanding, you really need to drop the 1899 version and recognize how the quantization of black body radiation satisfies the entropy definition, and recognize that, "... Even in thermal equilibrium, transitions associated with the absorption and emission of photons are occurring continuously... " as said by Einstein.