The invention proceeded no further in China, beyond incendiaries, fire lances, and firecrackers. European gunpowder and cannon were reintroduced to China under the Ming dynasty by the Portuguese and others.
This is clearly mistaken. The Chinese used cannons and handguns during the period of Mongol rule, as seen from surviving late-13th and early-14th century examples. However, Chinese cannons and handguns did not undergo the same rapid improvements that European and Turkish ones did in the 1400s and 1500s, for various reasons - the most important one probably being that the Ming Empire mostly waged war with the Mongols and the hill tribes of southwestern China, neither of whom was equipped with firearms. When Ming armies began clashing with arquebus-armed pirates and culverin-armed Portuguese warships in the early 1500s, however, they quickly began arming their own forces with these weapons.
Given the enormous impact that firearms (cannon and small arms) had on warfare, I tend to be skeptical of claims of Chinese priority in the invention of these specific things.
The problem here is that cannon appear in Europe and in China at about the same time, i.e. the early 1300s. The point of origin and direction of the spread of cannon technology are difficult to establish in the absence of textual evidence, but the likelihood that the Mongol Empire played a major role in the spread is very high. The recently discovered Chinese/Mongol handgun of 1271 (mentioned earlier in this thread) seems to indicate that hand cannons were found in China not long after they are first attested in Egypt in 1260, even if one discounts (as some have) Joseph Needham's dating of a Chinese sculpture with a 'hand cannon' to the 1100s or doubts the identification of the object depicted as a 'hand cannon'.