= Solution
At fixed external <momenta>, fixed regulator and fixed bare <coupling>, define the total scale derivative
$$
\mathcal D=\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial g}.
$$
The bare vertex has no independent dependence on the arbitrary <renormalization scale>. Differentiating $\Gamma^{(n)}=Z_\phi^{n/2}\Gamma_0^{(n)}$ therefore gives
$$
\mathcal D\Gamma^{(n)}=\frac n2\left(\mu\frac{d\log Z_\phi}{d\mu}\right)_{\mathrm{bare}}\Gamma^{(n)}.
$$
Defining the two scale-response functions as in the source yields
$$
\boxed{\left[\mu\partial_\mu+\beta(g)\partial_g-\frac n2\gamma(g)\right]\Gamma^{(n)}=0.}
$$
This is the <Callan-Symanzik equation>. Its minus sign and factor $n/2$ follow from the specified $Z_\phi^{n/2}$ multiplying the bare vertex. In particular, the source's $\gamma=\mu\,d\log Z_\phi/d\mu$ is twice the <field> <anomalous dimension> defined using $\log Z_\phi^{1/2}$. A convention change must be made consistently in both definitions and equation.
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