In the book of Shklovski (in russian) also the proof of this theorem is realized, based on the assumption that two distinct potentials can correspond to the same density. The assumption leads to a contradiction, so it is incorrect. In this proof, a direct correspondence is established between the density and the external potential. In your interpretation, the feature of the relationships between the wave function and electron density is subtly noted. Your interpretation of the proof of the theorem is very interesting.