There has always been a debate over this issue ... always will be.
Traction is all about contact patch and tread design. That is why, those who wheel, air down to increase contact patch. They also choose a tread design that best suits the terrain they want the most traction on.
Anyone who thinks that "Farmers know thinner tires are better", should not speak for all Farmers. Most of the time, they are wide in order to maintain a bigger contact patch and to reduce soil compaction. Ever seen dual tires on one? Most of the time, sinking down to solid ground is not much of a traction advantage plus it ruts the ground. "Tread Lightly"? Tractor tires usually have an aggressive tread design suited for the terrain.
Bulldozers also rely on contact patch and tread design. They too take advantage of a wider track in order to acheive this.
Snow and Ice seldom squeegees out from under a tire. Again, contact patch and tread design. Rain does squeegee out from under a tire, but relies on tread design to increase the contact patch.
Tires come in a variety of width and tread designs. Pick one that best suites your needs. Choose a tread design that best suits the terrain you will mostly be on. Pay attention to how the tire is rated for the terrain you mostly see. Usually, they are rated on a poor to good scale. If gas milage is a concern, go only as wide as you need to maintain a good contact patch for traction.
Les