Concerning Chinese cannons of the 1500s-1600s:
Apparently, the name "Great General Cannon"
dàjiàngjūnpào (大將軍砲) was a general term for a large cannon. Earlier on, they looked like the cannon in the first picture (there were also names like First and Second General, but I'm not sure exactly what they referred to).
EDIT: This picture won't show up for me (I don't know about anybody else), but I couldn't find another good picture of a Chinese Great General cannon.
The one below is a Great General cannon, but it's at a Korean museum, so I don't know how different it is from the Chinese version (I don't think much if really at all).
Since the name "Great General" was a general term, it sometimes applied to large breechloaders such as this.
After the introduction of the "Red Barbarian Cannon"
hóngyīpào (紅衣炮) around 1630 from the Netherlands, the Great General cannons began to be tremendously influence by them, and came to look more like the cannon in the third picture. Some of the older, pre-European Great Generals may have still been used on a small scale.
Small breechloaders (originally from Portugal) were called "Frankish (cannons)"
fúlángjī (佛狼機).
There was also a mortar-like cannon called the "Crouching Tiger"
hǔdūnpào (虎蹲砲).