A Question about the Cumulative Hierarchy.

My favorite definition of $V_\alpha$ (which I think ought to be standard) is the recursion that you quote from Wikipedia: $V_\alpha=\bigcup_{\beta<\alpha}\mathcal P(V_\beta)$. Notice that this immediately implies that $V_0=\varnothing$ (because there is no $\beta<0$) and that if $\alpha\leq\gamma$ then $V_\alpha \subseteq V_\gamma$ and, as a consequence, $\mathcal P(V_\alpha)\subseteq\mathcal P(V_\gamma)$.

In view of these facts, the definition also gives that $V_{\alpha+1}=\bigcup_{\beta\leq\alpha}\mathcal P(V_\beta)=\mathcal P(V_\alpha)$. So we have the $0$ and successor clauses of the definition you quoted from Devlin.

Finally, for a limit ordinal $\alpha$, we have (using the fact that the successor ordinals $<\alpha$ are cofinal among all the ordinals $<\alpha$) that $V_\alpha=\bigcup_{\beta<\alpha}\mathcal P(V_\beta) =\bigcup_{\beta<\alpha}V_{\beta+1}=\bigcup_{\beta<\alpha}V_\beta$, which is formula (3) in the question.