So if I run a large chess forum, and want to delete posts that aren't about chess, I would have to choose "publisher"?
No. Any such law could easily have provisions that exclude moderation for off-topic content (which would be defined carefully so as to include spam).
It would mean you can't delete "trolling", "bullying", on-topic flamewars and other categories of things you might want. To do that you'd have to indeed become a publisher. And then you'd need to respond to complaints quickly and take down libellous content, or become responsible for it.