I am not a student of the Chinese language (at the moment). However, this little fact, which I learned in a geography class long ago, occasionally keeps me awake at night.

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

You will also see some rivers in Fukien Province (and on Taiwan) labeled as Qi (Ch'i) but this is definitely a linguistic difference.

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:

I have an atlas but cannot find any resources that explain why there are three different words.  I can only speculate that they are the remnants of some long-forgotten cultural intermingling.

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".