You've missed a *key* word. "and use them *privately* in your own work and play". I don't think running a public web server can be classified as "private use".
And (forgive me if I'm wrong) I doubt the AGPL contains a requirement to notify anybody other than your users that you're using a modified version (and the ordinary GPL requires that, too, in the sense that anybody to whom you distribute a binary is a user).
So, I'm afraid I'd agree with RMS. And I don't think he's modified his stance, either. If somebody provides me with binaries based on GPL sources, they have to tell me that it's based on GPL code, and they have to make that code available to me. The AGPL is in principle exactly the same - if they provide a web service based on AGPL code, then they have to tell me that it's based on AGPL code, and they have to make that code available to me.
The only exception to this (which also applies equally to GPL and AGPL code), is that a company is considered "a user", so internal company use does not trigger the "must share the code" requirement, because an employee is not a user in the legal sense.