I was puzzled by the proof of $(i)\rightarrow (ii)$.
How to check that $E:M\rightarrow N$ is uniquely determined by the condition $\varphi\circ E=\varphi$?
To my mind, we need to verify the following statement: If $E'：M\rightarrow N$ is another conditional expectation such that $\varphi\circ E'=\varphi$, then $E= E'$.
But how to conclude the above conclusion according to the equality $\varphi(E(bx^*b))=\varphi(bx^*b)$?
The equality in 10.3, which is longer than what you wrote, says that $$ \langle \pi_\psi(E'(x))b_\psi,b_\psi\rangle=\langle \pi_\psi(E(x))b_\psi,b_\psi\rangle $$ for all $b$, and hence $\pi_\psi(E'(x))=\pi_\psi(E(x))$; as $\pi_\psi$ is faithful, $E'(x)=E(x)$.