Aren't your two formulations of a theory $T$ having DC equivalent? If DC is "true of $T,$" then $T$ proves $\phi$-DC by substituting $\psi(f, r):= (\forall g \exists s \phi(g, s)) \rightarrow \phi(f, r).$ This distinction only matters if we restrict DC to a fixed complexity of projective formulas, since $\psi$ has higher complexity than $\phi.$
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