I have some experience with classic car ‘tax free’ status. I currently have two exempt cars and yes both need to have had their taxation category changed to ‘historic’ in order to qualify for free tax.
The cars still need to be taxed but the cost is £0, in order to tax them they must first be insured.
They should also either have a current MOT or a self declared MOT exemption
The only exceptions to the above are if the car has been SORDed or has been off the road since before these regulations can into force.
I know of several people who had a tap on the shoulder for falling foul of the above regulations.