Yes, and presumably was originally “chains required” and “carry chains”.
The conflicting statements make me think there’s a fair amount missing. Probably trucks and vehicles over some specified axel weight must be actually using chains whereas cars & pickups merely need to have the chains with them in case conditions worsen.
That is, at least roughly, how CO does it. Tractor trailers were required to have chains (obviously not in August), cars were not required to have chains. Or even snow tires, but you had to have tires at least rated for snow (no summer tires) and they can check tread depth to make sure they aren’t too worn.