The key to the distinction lies in the idea that barbarians can become civilised. With regards to the contemporaneous Greeks, their attitude towards barbarians were that they were of sub-human capability unable to even become like the Greeks. For the Chinese, outsiders could be taught to act like the Chinese and thus become civilised. It was this attitude which perhaps caused invaders and usurpers of the celestial throne to eventually Sinify.
The Chinese attitude towards the "barbarian" peoples on its frontiers vacillated between "THREAT!" and disdain/pity depending on how Sinified said peoples were and how well the latest dynasty's "Northern Frontier Barbarian Management Program
(TM)" was going. That attitude, before modern times, never reached the proto-racism embodied by the Greek
barbaros, in which the barbarian was so inferior as to be unable to become civilized.
I've seen two etymological explanations for the Chinese
yeren (野人), which we traditionally translate into English as "barbarian". The first is "People from beyond the borders (of civilization); the second is "people of the earth". I cannot speak to which is more accurate in the original sense but both get a similar point across. I generally despise China-Rome analogies and avoid them like the plague but in this instance the Chinese concept of "barbarism" is generally compatible with the Roman one; in either case, through training, exposure, and proper behavior, a barbarian can transcend that status. In Rome, he could become a Roman citizen with all the rights and responsibilities of one born in the Eternal City itself; in China, a person could become Chinese, and whole nations could rise to a level of civilization very near to that of China itself as did Korea.
As for the Sinification of conquering peoples, insofar as I can tell, the attitude of the Chinese bureaucracy towards new ruling dynasties from outside China was that they would in the fullness of time become Chinese, their own customs and habits washed away by, or at least diluted in, the habits of the sea of Chinese over which they ruled. At no point did the bureaucratic establishment seem to have panicked and felt the need for a revolt, though the military establishment did so on occasion. The peasantry was generally less sanguine, as the dynamism of new rulers generally took the form of interfering in their lives more than end-stage "decadent" dynasties had done, but my reading of history suggests that this happened even when a "native" Chinese dynasty overthrew an "outsider" dynasty that had decayed. Certainly, the early Ming experienced no less chaos than the early Qing did.
It's also interesting that the Chinese policy for maintaining the security of their northern frontier, which was the central concern of every dynasty from the Sui to the Ming except the Yuan itself, rested on the principle of using the more sinified "barbarian" tribes to beat the crap out of the less Sinified ones without really ever raising arms themselves insofar as possible, then bribing all involved with wealth and a show of China's superior civilization. This policy seemed to have worked well so long as China's government was able to extract sufficient tax revenue from the truly massive economy under its control to pay for the necessary bribes, diplomatic visits, and armies. And so long as demographic and climatic factors didn't cause the barbarians to unify into tribal federations, which was apparently what produced the Jin, Mongol, and Manchu invasions.