Why the soundness error in the $\mathrm{IP}$ of GNI can implicate $\mathrm{\Sigma_2} \subseteq \mathrm{\Pi_2}$ if GNI is co-NP-Complete?

PDF here shows a way to proof GI is NP-Complete $ \implies \Sigma_2 = \Pi_2$ .

In the last step, it writes following:

In other words, (1) is false in this case as required.

Book Computational Complexity also follows the same way and supplies the $ \mathrm{\Sigma_2}$ formula:

$ \exists_{r \in \{0,1\}^m}\forall_{x \in \{0,1\}^n}\forall_{a \in \{0,1\}^{m`}}V(g(x),r,a) = 0$

It seems that if one wants to solve a problem in $ \Sigma_2$ , he can transform it into above form and call a $ \Pi_2$ machine to solve the contrary question and flip the reasult.

But in my opinion, Soundness error means verifier may accept the input which is not a member of GNI, so the $ \Sigma_2$ formula seems to be below and the error ratio is something about $ r$ rather than $ x$ :

$ \exists_{r \in \{0,1\}^m}\exists_{x \in \{0,1\}^n}\forall_{a \in \{0,1\}^{m`}}V(g(x),r,a)=1$

Is my opinion right?