I have always preferred Strahler. All the order 1 streams are headwaters, with no branching. One must realize that the branching evident can change with scale, with LiDAR yielding added detail to remote mapping efforts. Order 2, 3, etc. are additional orders as the branching continues downstream, with two order 1 to make order 2, two order 2 to make order 3, etc. If compiling or statistically comparing streams by order, all the orders would have a somewhat similar branching order pattern.

The Horton method assigns the longest stream the highest value of the stream or river order. So with a sixth order watershed, the longest segment is sixth order, from its headwaters to its outlet. The next longest major tributary would be 5 order, and this continues. Some may do this slightly different, and without accurate stream georeferencing of location, sinuousity, etc., lengths of channel distance might have some error. When comparing differences in stream order size, length, etc., the order 1 single branch headwaters would include one or more unbranched long tributary(s) with the higher values of long tributaries from the Horton value assigned to the longest tributaries.

The Shreve system is a little different as it counts the number of intersecting nodes as channel networks proceed downstream. The Shreve number is essentially the count of order 1 channels upstream of any location, based on the topographic scale of analysis, so there would be differences in 1:100,000, 1:24,000, and LiDAR based scales.

You might be interested in Stream Types and Management Implications on Researchgate as an example of how I used stream ordering in my career. Another example was the Loblolly Pine Removal Restoration Project in 2013.

Strahler is recommended.

Strahler is the better way. Read in detail why it is better.

I have preferred Strahler because it is straight forward and easy to follow.