Generally speaking, American men's hat styles in the 18th and 19th century reflected that of those in fashion in Europe, to include shape, color, materials, and decoration.

The rosette or cockade was a military feature to identify allegiance in an age when the primary color of the uniform did not yet identify the soldier's country of origin. For example, the French Army's Irish Picquets with Prince Charlie at Culloden wore red coats - though a different shade than that worn by the British line.

The cockade's shape was based on fashion of the day and, perhaps more importantly, the regimental commander's preference. Regimental commanders had a great deal of influence over the regiment's uniform features, as well as the arms and equipment carried by the regiment. In the British Army, for example, the colonel was given a budget to uniform and equip his regiment. Many colonels dipped into their own personal funds so that their regiments would be better uniformed and equipped, while others bought shoddy and lower priced uniforms and equipment, then pocketed the difference.

Cockades came in various shapes - from the extremely simple, perhaps consisting of no more than some ribbon shaped into an "X" or a circle to the extremely ornate, such as Robert's. Consequently, colonels could save money on the type of cockade chosen - from the quality of the ribbon to how much ribbon went into making it.

To make things even more diverse, officers could usually fancier items if they chose, so one officer might have a simple cockade, perhaps slightly larger and made with better ribbon than his men, while another might make one up like Robert's.

Civilians frequently picked up on the cockade, especially during wartime to show which side they supported (of course they sometimes had alternate cockades so they could switch them if necessary). This custom carried over into peacetime as well, allowing the wearer to dress up his hat a bit more, but began going out of fashion, generally speaking, after the Napoleonic Wars.

I note that the cockades on the current crop of Balmorals and Glengarries, tends to go more for the "bow tie" look.