A simple way to answer this question is: denote by $Spec$ the homotopy category of spectra, $Spec _0$ the category of (-1)-connected spectra, $InfL$ be the homotopy category of infinite loop spaces. The assertion that "connective spectra are the same as infinite loop space" simply means that the composition of functors $$Spec \to Spec _0 \stackrel{\Omega ^{\infty}}{\to} InfL$$ is an equivalence.

A simple way to answer this question is: denote by $Spec$ the homotopy category of spectra, $Spec _0$ the category of (-1)-connected spectra, $InfL$ be the homotopy category of infinite loop spaces. The assertion that "connective spectra are the same as infinite loop space" simply means that the composition of functors $$Spec _0 \stackrel{i}{\to} Spec\stackrel{\Omega ^{\infty}}{\to} InfL$$ is an equivalence, where $i$ denotes the inclusion. You gave an example of infinite loop space in the image of $\Omega ^{\infty}$, which doesn't mean that it is not in the image of $\Omega ^{\infty}i$.