In the UK the tradition for many club riders was to use a fixie over winter but one with clearance for wider tyres and fenders. It is simple, nothing to go wrong and is good for training legs. We dont get much deep, heavy snow but people ride fixies in the 3-4" that we do get. Your style of winter bike really should reflect the kind of winter you have.
It was not the only kind of winter bike, some people used a hub geared racer style bike or ordinary derailler race bikes.