To keep it simple I'll phrase everything in terms of scalar fields. We seem to have three constants called $Z$:
When we do LSZ reduction we say, as $t\rightarrow-\infty$ then $\phi\rightarrow \sqrt{Z}\phi_{free}$.
Later we renormalised the theory and rescaled the bare fields $\phi_0$ as $\phi_0=\sqrt{Z}\phi_R$.
We also have a factor $Z$ that is the residue of the two-point correlation function $\frac{Z}{p^2-m^2_R}$.
Why are these all the same $Z$? (ok, i sort of see that for 2 we can choose to rescale the field however we want, but that doesn't explain why 1 and 3 are the same)