(Note: As anyone can tell you, I am not an AGPL fan, but in this case, the poster above you is correct).
You want section 13, where it clearly states:
"Notwithstanding any other provision of this License, , if you *modify* the Program, your modified version must prominently offer all users interacting with it remotely through a computer network ..."
Nowhere else will you find anything related to releasing source due to network interaction.
It is only triggered if you modify the program first.