What is, diagrammatically, the 2-vertex $\Gamma^{(2)}$? I know that the 2-vertex $\Gamma^{(2)}$ is the second derivative of the effective action, but I fail to see what it is diagrammatically: is it the truncated 1PI diagram? The non-truncated one?
If this helps, the trouble comes from the identity, in massless $\lambda\phi^4$ theory, that states
$$\Gamma^{(2)}=G^{-1}=P^{-1}-\Sigma\tag{1}$$
where $G$ is the propagator, $P$ is the free propagator, and $\Sigma$ is the self-energy. I understand why
$$G^{-1}=P^{-1}-\Sigma\tag{2}$$
(using the geometric series), but I fail to see what
$$\Gamma^{(2)}=G^{-1}\tag{3}$$
means in terms of diagrams.
 A: The 1PI effective action $\Gamma[\phi]$ generates the 1PI connected amputated Green's functions. Let us denote the connected fully resummed non-amputated vertices by shaded circles, and the 1PI amputated vertices by empty circles.
Diagrammatically:
Note that the row for $n=2$ is $G = G\Gamma^{(2)}G$, which is the same as $\Gamma^{(2)} = G^{-1}$.
If you think about it, this is just drawing what amputated means: $\Gamma^{(n)}$ is the fully resummed vertex with all the fully resummed propagators $G$ removed from the legs.
A: *

*OP is correct that the diagrammatic interpretation of the self-energy $\Sigma$ as (a sum of) amputated diagrams comes from the geometric series (2), cf. e.g. this Phys.SE post.


*OP is also correct that the diagrammatic interpretation of eq. (3) is less clear. Eq. (3) is inherit from the Legendre transformation$^1$ between the generator $W_c[J]$ of connected diagrams and the effective action $\Gamma[\phi_{\rm cl}]$. Note that eq. (3) gets modified in the presence of tadpoles, cf. e.g. my related Phys.SE answer here.
References:

*

*S. Weinberg, Quantum Theory of Fields, Vol. 2, 1996; eq. (16.1.21).

--
$^1$ Eq. (3) often contains a minus, cf. e.g. Ref. 1.
