I asked chatGPT to explain the theorem and it said more or less the same thing you said. but it added an extra information. It said “The first Hohenberg-Kohn theorem states that the ground-state electron density ρ(r) of a many-electron system uniquely determines the external potential vext(r) acting on the electrons, up to an additive constant.” Here the extra part is the additive constant. i don’t know how to determine the additive constant.