Appears the twist change came in 1958 already, but the name change in 1960. Only 3 years of guns out there with the 244 moniker and the 1:12 twist, and a few more years of 244 with the faster twist, according to this (this info varies from Remington's site info by a few years):
6mm Remington (244 Remington)
It has been said on more than one occasion that when Remington engineers were developing the .244, their primary objective was to come up with a high performance varmint cartridge and for this reason decided to go with a maximum bullet weight of 90 grains. This isn't exactly true. Instead, after extensive testing of many bullet weights, it was decided that optimum weights for the .257 Roberts case necked down to 6mm were 75 grains for varmints and 90 grains for deer. A rifling pitch rate of 1-12" proved to be the best compromise. The rest, as they say, is history.
For reasons that no one can explain (or prove), deer hunters decided that the .243 Winchester with its 10 more grains a bullet was a better choice. Or at least this is the traditional explanation. Possibly a factor even of greater influence was the rifles the two cartridges were available in. The .244 was offered in the Model 722, a rifle about as handsome as a fence post. The .243 was available in the sexy little Model 70 Featherweight. When it came to looks, there was no contest.
Around 1958, Remington started rifling all .244" caliber barrels with the 1-9" twist, but nobody seemed to notice. In 1962 when the new Model 700 rifle was introduced, the only 6mm cartridge it was available in was the .243 Winchester, but during the next year the .244's name changed to 6mm Remington and reintroduced in the Remington bolt action.
While Model 722 and 725 rifles with 1-12" twists will stabilize 100 and 105 grain round nose bullets, they usually prefer 95 grains or less when loaded with spitzers. Since Remington dropped the 90 grain 6mm factory load, those who hunt deer with the early 722's have no choice but to handload. Later .244 Remington rifles with 1-9" twist will handle the 100 grain factory load.
Source: Hodgdon Data Manual, 26th Edition
FWIW