Read more carefully. No offense, but you missed a lot that's specifically spelled out in the statutory language.
30-7-2.1 spells out schools- and does not limit it to just elementary schools- it is "schools" in general.
30-7-2.4 pertains specifically to universities.
In both statutes, the definition of premises includes playing fields and areas in which "sanctioned activities are being performed".
All of which means- NO, you cannot CC or carry at all.