Geography Question

A river flows downhill from a mountain, a spring, or from a lake.  The beginning of a river is called its "headwater," and its ending is called its "mouth."  Along the way, it meets up with another river, flowing downhill from another source.  The two rivers combine at that point, called a "confluence," and thereafter the downflow is in a single river.

Sometimes, both contributing rivers end, and the resulting river is given a new name.  For instance, at Pittsburgh, Pennsylvania, the Monongahela River and the Allegheney River meet, and their confluence is the origin of the Ohio River.  Thus, Pittsburgh is the mouth of the Monongahela, the mouth of the Allegheney, and the headwater of the Ohio.

But sometimes, one river is called a "tributary" of another river. For instance, at Cairo, Illinois, the Ohio River meets the Mississipi River, and the river that flows downhill from the confluence at Cairo is called the Mississippi. Cairo is the mouth of the Ohio.  But it is merely another spot along the course of the Mississippi.  Go north-by-northwest about 140 miles from Cairo, Illinois to just north of St. Louis, Missouri, and you will find the spot where the Missouri River becomes a tributary of the Mississippi.  Some geographers refer to that portion of the Mississippi north of these confluences as the "upper Mississippi" and the rest of it as the "lower Mississippi" but that does not seem to be a universal appellation.
 
So why is it that the confluence of the Ohio and the Mississippi produces the Mississippi River, while the confluence of the Monongahela and the Allegehney produce the Ohio?  How does a geographer or a hydrologist distinguish a situation in which one river "takes over" a tributary from a situation in which two rivers unite to create a third?  Or is it just arbitrary?  Inquiring minds want to know.