I just came back from a long wikibreak, and it became suddenly very apparent that the ability to mark *all* notifications as read in one click is not available in the JavaScript free version of Special:Notifications. (Unless I'm missing something?)
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There is no solution at the moment to mark all discussions as read unless by using JavaScript. The task has just been claimed, I hope some changes will happen soon.
This is now done, thanks to @MtDu. It'll roll out next week (Aug 15-17, depending on the wiki)
This feature has now rolled out on all wikis.