I think you're getting a bit confused there. AGPL doesn't so much place restrictions on the user, but instead redefines who the user is - the person who interacts with the software rather than the person who owns the server that it is running on. If you don't modify AGPL code, then you're fine.
Not "obeying" the GPL doesn't alter the license of the code at all. It's simply that the GPL can grant distribution rights if you abide by it and if you don't abide by the GPL, then you no longer have any distribution rights to it and can be considered to be violating copyright (i.e. unauthorised distribution).