Exact propagator - 1PI diagrams

Above diagram can be written in terms of series:

$$i\Delta = -\frac{i}{p^2 + m^2} + \Big(-\frac{i}{p^2 + m^2}\Big)(i\Pi)\Big(-\frac{i}{p^2 + m^2}\Big)+ \Big(-\frac{i}{p^2 + m^2}\Big)(i\Pi)\Big(-\frac{i}{p^2 + m^2}\Big)(i\Pi)\Big(-\frac{i}{p^2 + m^2}\Big) + \cdot \cdot \cdot = - \frac{i}{p^2 + m^2 - \Pi}.$$

So to get the exact propagator, I should sum over all the 1PI diagrams. 1PI diagrams are those from which I can't get two diagrams by cutting a line. But then $$3$$rd diagram and those arising from the higher order terms are really 1PI diagrams? (since I can cut the $$3$$rd diagram and obtain two diagrams)

1. To get the full propagator/exact propagator/connected 2pt function $$\Delta_c~=~\Delta_0\sum_{n=0}^{\infty}(\Pi \Delta_0)^n\tag{A}$$ one should sum over self-energy diagrams $$\Pi$$ and free propagators $$\Delta_0$$, cf. e.g. this Phys.SE post.
2. Since the free propagator $$\Delta_0$$ is not a 1PI diagram (it can be cut in 2 by a single cut) none of the terms on the RHS of eq. (A) are 1PI diagrams, cf. OP's question.