No, this is an AGPL misconception that won't die. What it actually does is impose a restriction on *modification*.
Specifically, it says that if you modify the software you have to *cause* it to make its source code available or offered to network users. You are not allowed to modify the software without making *that* modification first.
Absolutely nobody complies with this properly.
Nothing in the AGPL alters the distribution terms whatsoever. The FSF did it like this because they know if they did that, it would be blatantly non-FOSS.
Alas, this goes against the spirit of the OSI definition #3 (it restricts derived works). So it still should have never been called FOSS.