QED mass counterterm coefficients and the bare mass ($\delta m_L=Z_2{m}_0-m$)

With renormalized Dirac fields, the coefficient of $-\bar{\psi }\psi$ in the counterterm Lagrangian is $\delta m_L=Z_2{m}_0-m$, whereas the bare-parameter shift is $m_0-m=\delta m_L-m\delta Z_2$ at first order. In Feynman gauge and $d=4-ϵ$, QED has $\delta Z_2=-e^2/(8{\pi }^2ϵ)$ and $\delta m_L=-4m{e}^2/(8{\pi }^2ϵ)$. Their difference gives $m_0-m=-3m{e}^2/(8{\pi }^2ϵ)$. Defining the insertion and the mass coefficient before quoting its sign avoids confusing these two counterterms.