Auto insurance companies, rarely if ever, are concerned about reasons and justifications. They simply focus on the facts and draw their own conclusions. If you overshot the red light and hit another vehicle, your insurance company will focus on the amount it had to pay on your policy. It will not take into consideration that your tires did not grip the road and you skidded through the red light. The company would merely insist that the job of maintaining the tires and preparing it for the cold season too was your own responsibility.
If you want to avoid being stuck in silly mishaps which may lead to huge increase in your insurance premium, you should take steps to prepare your car for the winter season.