The GPL is a copyright license. It applies when copyright law would otherwise prevent distributing some program. If I were to violate the GPL, for example by sending someone a compiled version and refusing to provide the source, then there's a clear legal theory by which copyright law could be applied.
That "one added requirement" in the AGPL turns it into a EULA, because it's an attempt to regulate for which purposes a user may run the program. If I were to run a modified AGPL'd SSH server on my own hardware, the question of whether I'm violating the AGPL depends on whether I'm allowing others to access it remotely. If the AGPL can be violated without copyright infringement, it's clearly in a different legal category than the GPL.