Here is a definition of holomorphic convexity taken from the notes of Eyssidieux:
Defintion. A complex analytic space $S$ is holomorphically convex if there is a proper holomorphic morphism $\pi: S\to T$ with $\pi_*O_S=O_T$ such that $T$ is a Stein space. $T$ is then called Cartan-Remmert reduction of $S$.
Questions. 1) Is this correct that Cartan-Remmert reduction is unique is if exists?
2) Do I understand correctly, that (assuming properness of $\pi$) $\pi_*O_S=O_T$ just means that $\pi$ is a surjective map and all its fibers are connected?