
Hah! I’m always happy to “muddy the waters” of a disagreement. (See what I did there?)
Unlike the simple rules of the Thunderdome (“Two men enter, one man leaves!”), there’s no single rule for labeling streams.
Everybody agrees that a stream with NO tributaries is a “first-order stream.” And most people agree that when two first-order streams merge, they deserve to be called a “second-order stream.”
That’s about where the agreement stops.
There are two different methods whipped up in the 1950s and 1960s that are still in use today for ordering streams:
The Strahler Method: Developed by Arthur Strahler, this method says that stream order only increases when two streams of the same order merge. If a lesser stream joins a greater stream, then the stream order doesn’t change. (For example, two second-order streams merge to make a third-order stream; but if a second-order stream joins a third-order stream, it remains a third-order stream.)

The Shreve Method: Developed by Ronald Shreve, this method takes the approach of adding the order numbers of any streams that merge. Every time there’s a merger, the order number changes. (For example, if a second-order stream joined a third-order stream, the result would be a fifth-order stream.)

There’s no single, correct method. Lots of hydrology planners prefer the Shreve method; it seems to be the more popular method as of 2021.
However, the Shreve method has the advantage of telling you how many upstream channels there are at a glance; some watershed-based maps prefer this method.
Regardless, the folks in the GIS and mapping industry are ready to switch between methods when necessary. For example, the ESRI company has a switch to let you choose which method to use for your map.
(The streams, presumably, don’t care what humans label them; they just flow.)
-Naturalist Chris Mentrek