I know that Einstein introduced his cosmological constant assuming it as an independent parameter, something characteristic of the Universe, in itself, but the term of it in the field equations can be moved "to the other side" of equality, written as a component of the energy-tension tensor $T$ for vacuum:
$$T_{\mu\nu}^{(vac)}=-\cfrac{\Lambda c^4}{8\pi G}g_{\mu\nu}.$$
Since this result would correspond directly to the energy density $ρ$ in the energy-tension tensor $T$, this has the inevitable consequence that we are already talking about the vacuum energy given by the following relationship according to General Relativity:
$$\rho_{vac}=\cfrac{\Lambda c^2}{8\pi G}.$$
Thus, the existence of a cosmological constant Λ different from zero is equivalent to the existence of a vacuum energy different from zero; there is no way in which we can escape this conclusion. This is why the terms cosmological constant and vacuum energy are used interchangeably in General Relativity.
But I do not understand if it is correct to say that the cosmological constant and the energy density of the vacuum have the same value or how to prove that they actually have the same value, could someone help me?