That's not quite as I recall things. The original 1960s "off-peak" system provided a time-switched storage heater supply via a separate meter, but this was not referred to as a white meter.
"White meter tariff" came along toward the end of the 60s and was the same principle that we now call Economy 7, i.e. a 2-rate meter and an optional switched supply for storage heaters.
Which of course they did. That's probably one reason why white meter / E7 was devised.
You can still get, I think, "Economy 10" tariff from some suppliers, giving something like 00:00 - 05:00, 13:00 - 16:00, and 20:00 - 22:00 at a cheaper rate (but not as cheap as the E7 night rate).