I have always seen it as the vehicle that is insured to be driven by named drivers, this is why you need a policy covering each vehicle. In most cases the vehicles are specified in the policy, or you can get a policy covering any driver for a specific car. A motor traders policy is basicly intended to enable them to run a business.
The "driving other vehicles" is a bonus on some fully comp policies, you were obviously aware that you had it, and carried it with you. When the police, quite correctly, stopped you, you were able to produce it, so whats the problem?