Greg Martin is right: Maier's claim is in error. Precisely, the following can be said (cf. Corollaries 11.15 and 11.20 in Montgomery-Vaughan: Multiplicative number theory I):

**Theorem (Siegel-Walfisz with a twist).** Let $\epsilon>0$ be fixed. Let $q\geq q_0(\epsilon)$ be sufficiently large in terms of $\epsilon$. There is an absolute constant $c>0$ such that
$$\theta(x,q,a)=\frac{x}{\varphi(q)}+O(x^{1-cq^{-\epsilon}})\qquad\text{for}\qquad x\leq e^{q^{2\epsilon}},$$
and
$$\theta(x,q,a)=\frac{x}{\varphi(q)}+O(x e^{-c\sqrt{\log x}})\qquad\text{for}\qquad x>e^{q^{2\epsilon}}.$$
The implied constants are absolute.

Perhaps Maier only needs the range $x\leq\exp(q^{2\epsilon})$, so his final conclusions are OK.