A principal-value sum (or integral) is usually one in which unconditional summation (or integration) does not converge, so one needs to sum in a particular way to achieve convergence. I suspect that, in this case, the necessary summation is symmetric, so that we consider $\lim_{N \to \infty} \sum_{n = -N}^{n = N} f(n)$ instead of $\sum_{n = 1}^\infty f(-n) + \sum_{n = 0}^\infty f(n)$.
It's not quite clear to me what your issue is with the two formulæ you mention. Since you are summing different functions ($1/(z + n)$ versus $z/(z + n)$), it is no surprise that the answers are different. What am I missing? (Sorry, I did not notice that you had already factored out the $z$.)
A principal-value sum (or integral) is usually one in which unconditional summation (or integration) does not converge, so one needs to sum in a particular way to achieve convergence. I suspect that, in this case, the necessary summation is symmetric, so that we consider $\lim_{N \to \infty} \sum_{n = -N}^{n = N} f(n)$ instead of $\sum_{n = 1}^\infty f(-n) + \sum_{n = 0}^\infty f(n)$.
It's not quite clear to me what your issue is with the two formulæ you mention. Since you are summing different functions ($1/(z + n)$ versus $z/(z + n)$), it is no surprise that the answers are different. What am I missing?