The original "White Meter" set up was this: -
A standard meter with a Voltage and a current coil,
two dials selectable by a small solenoid inside.
An additional live output terminal
An additional switching terminal to operate the solenoid.
Also as part of the system was a time switch set to the appropriate times applicable to the tariff in use (this was later changed to a radio teleswitch in some areas to allow more flexible times.
The connection set up was this
L & N from cutout to meter,
L1 output to on peak supplies
L2 output to time switch
2 neutrals possibly via a connector to the two supplies and the timeswitch
Input from switched terminal of timeswitch
From the timeswitch the L then continued to the off-peak supplies (usually storage heaters and immersion heater)
In this way all load was metered at either the on or off peak rates.
When the off peak switched in the load was controlled by the timeswitch (it had to be as the distribution system design was made to cater for this and not for the load switching on at random times set by the customer/electrician)(it still should be BTW)
The off peak load switched on and the solenoid in the meter changed the drive to the off peak dial, all load was metered at the off peak rate, hence the sales talk of operating automatic washing machines, dryers etc at night on the off peak rate.
Why was it a white meter, it was purely a sales gimmick, it could have been red or blue or any other colour than black!
Other tariffs used two meters with all on peak load being metered at that rate and a separate off peak meter only connected to the off peak loads via the time switch.
All modern dual rate meters still operate to this system and often incorporate the radio switch, so again all off-peak load should be connected to the switched output.
You also should not that a meter does not need the N in and out terminals connected to operate correctly, it just needs a single neutral, the second terminal is just for convenience