= Solution
Fix the sign convention by taking the proper two-point insertion to be $i\Sigma(\not p)$. This convention matches the loop expression and mass conversion printed later. <Dyson resummation> of successive <fermion self-energy> insertions gives
$$
iG=\frac{i}{\not p-m}+\frac{i}{\not p-m}(i\Sigma)\frac{i}{\not p-m}+\cdots
=\boxed{\frac{i}{\not p-m+\Sigma_R(\not p)}}.
$$
The subscript denotes the renormalized <self-energy>. If the insertion is instead named $-i\Sigma_{\rm usual}$, then $\Sigma_{\rm usual}=-\Sigma_R$ and the denominator is written $\not p-m-\Sigma_{\rm usual}$. These are the same physical convention.
<Lorentz covariance> allows $\Sigma_R=\mathcal A(p^2)\not p+\mathcal B(p^2)m$. The physical mass-shell condition is
$$
\boxed{[1+\mathcal A(m_{\rm phys}^2)]m_{\rm phys}
-[1-\mathcal B(m_{\rm phys}^2)]m=0.}
$$
It locates the mass-shell singularity of the <Dirac propagator>. In infrared-regulated perturbation theory this is the <pole mass>; near a simple pole the <quantum field theory propagator> has the form $iZ_{\rm pole}(\not p+m_{\rm phys})/(p^2-m_{\rm phys}^2+i0)$. The mass and pole residue are different quantities: the former fixes the singularity's location, the latter the field normalization. The calculation below uses this standard perturbative pole-mass definition.
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