When searching for the Bombieri-Vinogradov theorem on the internet, it appears under different forms, but with some minor differences there are essentially these two: $$ \sum_{q\leq Q}\max_{(a,q)=1}\left|\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\right|\ll_A\frac{x}{(\log x)^A}\tag{pi}\label{pi} $$ and $$ \sum_{q\leq Q}\max_{(a,q)=1}\left|\psi(x;q,a)-\frac{x}{\varphi(q)}\right|\ll_A\frac{x}{(\log x)^A}\tag{psi}\label{psi} $$ where $A>0$ is arbitrary and $Q=Q(x,A)=\sqrt{x}(\log x)^{-B}$ with $B=B(A)$ sufficiently large depending on $A$.
Furthermore, there seems to be no mention of there beeing these two different forms; it is always stated "this is the Bombieri-Vinogradov theorem" and then a slight variation of either $\eqref{pi}$ or $\eqref{psi}$ is shown. Is the transition between the two really so trivial that it isn't even worth to tell them apart? Because I fail to see an elementary argument proving their equivalence (or even for one implying the other). So what exactly is the relation between $\eqref{pi}$ or $\eqref{psi}$?