You have the idea of fast and slow twist backwards. A 1:10" twist (one turn in 10") is "faster" than a 1:12" twist (one turn in 12"). A fast twist spins the bullet faster.

The original version of Remington riflres for the 6 mm Rem (called the .244 Rem at the time) had a 1:12" twist and it was too "slow" to stabilize bullets heavier than about 90 gr. Remington thought it was going to be just a varmint round and wouldn't need heavy bullets.

The .243 Winchester was seen as a dual purpose right from its introduction (varmint and deer size big game) and came with a 1:10" twist (i.e. a "faster" twist) that would stabilize 100 gr and even slightly heavier bullets. The .243Win was a tremendous success right from the start and the .244 Rem was a VERY slow seller.

Remington saw the error of its ways and changed their rifling twist to, IIRC, 1:9" and renamed the cartridge the 6 mm Rem to try to divorce it from the .244's poor reputation.

So, yes, a 1:10" twist will be fine for a 6 mm Rem unless you plan to use very heavy bullets or very long for weight target bullets like the "VLD" type. Then an even faster twist would be better.