The effective 1PI action $\Gamma$ generates the sum of all *proper* diagrams, which are:

 1. Connected
 2. Truncated
 3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the *exact* inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$
Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)). In fact, for $n=2$, sticking to the above recipe gives: $$G^{(2)}_{\text {trunc}} (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1}=\Gamma_2(p),$$
so that $G^{(2)}_{\text {trunc}}$ is in fact just the proper (1PI) function.
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The connection of truncated functions with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$
A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$
where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

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[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.