The Klingon-like Klingons are probably a separate race. Perhaps they were the ones who had been living on the planet before the Klingon homeworld was destroyed. This would explain why they retained some of the subtle features of the original race, such as the top of the head. However, it would also mean that the Klingons on the homeworld are related to them, not to the Federation.
Other explanations for the Klingon-like Klingons have also been suggested. They could be aliens from another galaxy that learned how to build/design spaceships. Some hypothesized that the Klingon-like Klingons were manufactured by the Romulans, but they are not consistent with the fact that some (but not all) Klingons are supposed to have been created by duplicating the genetic material from a Romulan. They could also be a lost civilization, which tried to live on the Klingon homeworld. The Klingon-like Klingons could also be the descendants of colonists or settlers who found a planet which resembled a Klingon world. If such settlers or colonists had been living for a long time on their new planet it is at least possible that their genetic material could have been altered. A more distant explanation is that the Klingons on the Klingon homeworld were a kind of cosmic migrants. They could have somehow arrived on the planet and have been assimilated into the native population. Perhaps some of them were assimilated, others changed their appearance and went away to found their own Klingon culture. Again, this is an explanation without a need of a parallel race.
Maybe the smooth Klingons are the descendants of the Klingons on Earth who were discovered by the original Klingon-Federation war. There was a rumor that the fabled Klingon homeworld was located in our galaxy.
The two possibilities of Klingons being civilized or savage do not contradict each other. The original Klingons were fierce savages. They were descendants of the Klingons who came from the Klingon homeworld. The smooth Klingons were descendants of the original Klingons who were civilized.
The most likely explanation for the smooth Klingons of TOS is that they were originally intended to be only a kind of special task force to deal with the Federation. These Klingons were developed to be more human-like, but were given a slightly different genetic makeup. When this special task force failed, they were all killed by the Klingons on their planet (and all the planets in the system), which led to a completely different genetic makeup. After leaving their homeworld, they drifted into the Federation. They are then used as special task force again to deal with the Federation, and those who failed are killed by the Klingons again.
There is also the question if the greys' appearance changed over the years. The black-haired Klingons are the ones who have always lived on the Klingon homeworld. But if so, why are the black-haired Klingons all from the same era? Why are the greys older than the black-haired ones? Why did the greys on the Enterprise-D look older than the ones in the Enterprise-E? 827ec27edc