Can a measurement commuting with the stabilizer of a state disturb the state?
The quoted claim from the book is true. It is not true that $Z_1$ must always commute with all stabilizers of $\lvert \psi \rangle$. For example, if $X_1X_2$ stabilizes $\lvert \psi \rangle$ (e.g., if $\lvert \psi \rangle$ was an EPR pair) then $Z_1$ does not commute with $X_1X_2$ and $Z_1$ is not a stabilizer of $\lvert \psi \rangle$.