Best winter tire size

[SECProto]

Not sure if this should be in the Tires section, or the 3rd gen section - but I put it here because it is a question specific to the 3rd gen Yaris.

I've had a 3rd gen Yaris since July, with the stock steel rims and 175/65R15 tires. I'm looking to get a set of winter tires to deal with our Canadian winter - we had some snow this week, and I had a bit of traction loss on hills.

I'm wondering what tires and wheels I should go for? I think I can get a set of Michelin X Ice tires with another set of wheels in 165/65R15 (see here and here) - would this be my best option? or should I be switching to a different tire profile? I see a lot of people talking about 185 or 195 widths. Is there any advantage to these sizes in winter conditions? note that although the wheel picture shows a 5-bolt pattern, the specs says 4 bolt.

Another option is to get a set from the dealership - they offer a ContiWinterContact ts810s for $115 each ($35/tire cheaper than the michelin xi3) and wheels for $70 each ($20/wheel more expensive) - but I don't know if their performance on ice is a good as the xi3.

So those are the two options I've explored so far. Apologies for being one of the many noobs to ask tire questions - i did several searches and read quite a few posts, but most of the discussion has centred around older yarises, and installing tires for looks and speed performance, rather than winter condition performance. Any advice would be much appreciated!