User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-7484840-20130709093331/@comment-5529393-20130709110039

The difference is that the above definition of the $$\psi$$ function contains the $$\varphi$$ function among its closure functions, rather than just $$\alpha \mapsto \omega^\alpha$$.

$$\psi_{\Omega_{I+1}}(0)$$ is defined as the smallest ordinal $$\beta$$ such that $$C(0, \beta)$$ contains no ordinals between $$\beta$$ and $$\Omega_{I+1}$$. Since $$C(0, \beta)$$ contains I and is closed under + and $$\varphi$$, $$\beta$$ must be at least $$\Gamma_{I+1}$$, and one can see that $$\Gamma_{I+1}$$ works.