I can attest to this.
My Civic is FWD, but it has A/T tires and Discount Tire told me they would be fine in the snow.
WRONG. They blow.
Do winter tires have their own designation, or can they fall under A/T?
They can be A/T, LT or some other type of tire but in the US and Canada they also have to have the Apline Symbol.
It designates they have been tested for winter conditions.
Even with the symbol some are better than others on Snow and/or Ice.