> Or do I misunderstand something?

The AGPL is identical to the GPL, but with one additional clause: if you use the modified software to provide a networked service (like a website), then you must distribute your modifications. That's all.

You can use AGPL'd software to produce content without fear that that content will be encumbered, since you're not providing a networked service through that content. That's why your documents are safe, even in the (unlikely) event that your word processor is AGPL.