Though the light cavalry sword of 1960 was supposed to supplant the European make heavy cavalry sword of 1840, the earlier sword was much preferred through the war and for some time after.
The difference between light and heavy cavalry was mostly in application. Light cavalry was for reconnaissance, screening maneuvers, raids, pursuits and that sort of thing. Heavy cavalry was used for front breakthroughs and flank attacks to roll up an enemy front.
In many ways they functioned like chess pieces, and Napoleon and his generals brilliantly used them as such.
In this sense, light cavalry were used like bishops and heavy cavalry like knights. (Engineers were the rooks, still their symbol, Infantry was the Queen (”of battle”) and Artillery the King (”of battle”).
When looking at the US Civil War, it is important to examine the Napoleonic Wars as well as the Mexican American War. All the US and Confederate officers had studied the Napoleonic Wars, and many of them had participated in the Mexican American War.
Ironically, at about the same time as the US Civil War, and lasting much longer, was the enormous Taiping Rebellion in China, which is perhaps the second bloodiest war in human history, and dwarfed the US Civil War in many ways.