What is the right way to calculate the inverse Z-transform of $zX(z^{-1})$
The first result is correct, the second is wrong. You're right that $x_i[n]=x[n-1]$ corresponds to $z^{-1}X(z)$. And if you replace $z$ by $1/z$ you need to replace $n$ by $-n$. But $x_i[-n]$ is simply $x[-n-1]$,  and not $x[-n+1]$. Note that you don't replace the whole argument of $x[n-1]$ by its negative value, but you just replace $n$ by $-n$.