Chinese has three words for "river"

Chinese indeed has three different words for "river".  These are:

• 河 He (Ho).  These include the "Yellow River" Huang He, Wei He, Huai He, and the "Grand Canal", Da Yun He.
• 江 Jiang (Chiang, Kiang). The largest are the "Yangtze River" Ch'ang Jiang and the "Amur River" Heilongjiang, but there are many, many others.
• 水 Shui. The largest Shui is the Han Shui in Hubei Province.

There does not appear to be a rule for assigning one label to a particular river.  There are large, medium and small "Jiang"'s, "He"'s, and "Shui"'s all over China.  Now it appears that "He" is concentrated in Northern China, "Jiang" in Southern China, and "Shui" in Central China, but there are notable exceptions.  For instance:

• The aforementioned Heilongjiang forms China's northern border with Russia.
• The Ya-lu Jiang forms part of the border with North Korea.
• The Wan Chuan He is on Hainan Island, and you can't get further south than that.

A final puzzlement:  I wonder which of the three words has been assigned to individual rivers outside China. It would be really interesting to know whether the Mississippi River is a "He", a "Jiang" or a "Shui".

But Chinese literate culture has been using these terms somewhat indiscriminately in the many thousands of years since jiang1 was "foreign", and that's why the geographic distribution seems so hazy.

Shui3 and xi1 (qi1) are also common words, but they refer to smaller rivers. And various dialects have sometimes adopted special terms for creeks and brooks - I've heard "port" (kang2 ) and "dike" (pi1 ) used for local bodies of flowing water in Taiwanese, and I'm sure there are others. Kang2 really refers to a deep ditch that leads out to a nearby body of salt water, so that sea fish can sometimes be caught in it. Pi1 seems to refer to a river with a high embankment.

'He' referred exclusively to the Yellow River ('Huanghe' in modern Mandarin) and 'Jiang' exclusively to the Yangtse (called 'Changjiang' these days). Our word Yangtse comes from the name 'Yangzi' applied to one section of the upper reaches of the river in Sichuan if I remember correctly. These original meanings doubtless explain the geographical spread of different modern usages noted in the write-ups above - the Yellow River (aka China's Sorrow because of its frequent flooding) is in the north, the Yangtse in the south.

In modern Mandarin 'Jiangnan' still refers to the part of the Celestial Empire indicated by the literal meaning in Classical Chinese - China south of the Yangtse.

In answer to Gorgonzola's question about which term is used in translation of foreign rivers, it seems he is your winner - a quick flick through my dictionary shows it used for the Nile, Tiber, Thames, Amazon, St. Lawrence and Volga. Didn't find any usage of jiang - though it was hardly an exhaustive search. It's true for the Mississippi too - all together now 'Mixixibihe'.