Yes, the owners manual is wrong, the box is right, the gap on the 112/113 engine plugs is 1mm. I have noticed the owners manuals being wrong a time or two on tech info like this. The spec had been .8mm for so long some guy who is in charge of what goes into the owners manual probably refused to believe it was different!
This whole tool for removing the spark plug boots I knew would cause some confusion. I can try to remember to post the tool number. It's not a socket like what you posted a picture of. It's a wrench, it's 17mm open end wrench, probably around 12" long with 2 pretty tight angles, like for clearing obstacles. The open end of the wrench will engage into the "groove" in the metal end, then the handle of the wrench can be pried against the valve cover to pry the end off of the plug. Easier to do than describe. If I had a digital camera I would post a few pictures of how to do it. Maybe I'll ask for one for my B-day this year or something.