= Solution
{c}
In $d=4-2\epsilon$, the <modified minimal subtraction scheme> subtracts $\Delta_{\overline{\rm MS}}=1/\epsilon-\gamma_E+\log(4\pi)$ for each one-loop pole, with no further finite, kinematics-dependent subtraction. Equivalently, use a rescaled <renormalization scale> $\overline\mu^2=4\pi e^{-\gamma_E}\mu_R^2$ and subtract only $1/\epsilon$ in that convention. This differs from the <minimal subtraction scheme> by a fixed scale convention.
Let $C=e^2\Delta_{\overline{\rm MS}}/(16\pi^2)$. With the source's <counterterm> insertion signs, cancellation proceeds as follows. The vertex has $-ie\gamma^\mu C$ from the loop and $-ie\gamma^\mu\delta_1$ from its <counterterm>, giving $\delta_1=-C$. The kinetic part of the <fermion self-energy> insertion is $+i\not p C$ and its <counterterm> is $+i\not p\delta_2$, giving $\delta_2=-C$. Finally the <photon vacuum polarization> insertion is $-i(q^2g_{\mu\nu}-q_\mu q_\nu)(4C/3)$ and its <counterterm> is $-i(q^2g_{\mu\nu}-q_\mu q_\nu)\delta_3$, giving
$$
\boxed{\delta_1=\delta_2=-\frac{e^2}{16\pi^2}\Delta_{\overline{\rm MS}},\qquad
\delta_3=-\frac{e^2}{12\pi^2}\Delta_{\overline{\rm MS}}.}
$$
Writing $Z_i=1+\delta_i$ therefore verifies the <Ward identity for QED renormalization constants>, $Z_1=Z_2$, at one loop. The <proper-vertex Ward-Takahashi identity in QED> independently gives $\Lambda^\mu(p,p)=-\partial\Sigma(p)/\partial p_\mu$, whose divergent part reproduces the opposite signs of $F_1^{(1)}$ and $S$ found above. The equality holds in a common <gauge fixing> and subtraction prescription; the individual constants $Z_1,Z_2$ are gauge dependent.
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