When did hand cannons become handguns?
Answer: Actually, "hand-cannons" and "hand-guns" is used interchangeably. See Heilongjiang hand cannon, note "a" explains this point with reference to Joseph Needham's Science and Civilisation in China. Incidentally, the Xanadu Gun is also considered a hand-cannon.
If you mean to ask what came after hand-cannons, then I believe it is the arquebus.
Indeed, one of the fascinating points that emerges out of a global war- ring states perspective is that modernization— the systematic adoption of more advanced technologies and techniques— is not something that arrived suddenly in Asia in the 1800s. As other scholars have suggested, it’s a long, deep process. The first gunpowder weapons evolved in a process of mutual interadoption during a period of warfare in East Asia from 900 to 1300. They spread beyond East Asia— probably carried by warring Mongols and their allies— and took root in Europe by 1320 or so, where they evolved quickly, only to be reexported in turn. The Ming adopted Portuguese cannons in the early 1500s, Japanese and Portuguese arquebuses in the mid- 1500s, and advanced Western artillery in the 1600s. One scholar argues that China’s adoption of such artillery was China’s first “self- strengthening movement.”
Source: Andrade, Tonio (2016), The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History, Princeton University Press, p. 10.