Well, honestly, it becomes a more useful road bike. Thinking about it, there may not be discrete categories, rather there's a spectrum. 32mm tires push a bike further away from a pure road bike and towards the touring side. Same thing with longer chainstays, higher handlebars and racks. Fenders too.

I run Panaracer Urban Max 32c tires on my commuter. Not the liveliest riding tires available, but they provide good protection from road hazards, and enough traction for mild off-road riding. It's no fun trying to fix a puncture on the side of the road when it's dark and below freezing, so I'm willing to sacrifice ride quality for peace of mind. Last night's ride home included ~3 miles of dirt, about half of which was covered in light snow. The aforementioned Panaracer's made it seem almost easy.

You can't win unless there is a race.

There's always a race: Cat 6.

I'm having trouble seeing your bike, but I find 28-32 to be a nice all-around size. I'd give the stock tires a shot before making plans about changing them.