There are innumerable stories of the origin of the word cocktail. There's the Mexican Princess who made a cocktail for visitor to her father's court. Unfortunately, there doesn't seem to be a princess in the history of Mexico with a name remotely like Cocktail. Mexico did however have a long history of rum production when it was under Spanish rule. But that's a separate historical footnote.
Was it named after a type of race horse? A tavern keeper who stirred drinks with a feather? Was it from the barrel dregs? From cock ale? A garnish of mint? The foam that descended down the side of the glass? (These last two were suggested as drinkers pondered this question in the mid-19th century.)
Looking at the 1806 definition: a drink made with spirits of any kind, sugar, water, and bitters...vulgarly called a bittered sling. Yes, I'm paraphrasing here, just to get to the point.
A bittered sling. So what separated the cocktail from the sling was the bitters? Add bitters to a sling and it becomes a cocktail. So far so good. Now, let's take a closer look at the bitters bottle.
Some early dasher bottles were made by pushing a short length of feather shaft through a cork. Modern bitters dasher bottles are still made in this shape. The feather would have to be a large one to give a large enough diameter for dashing bitters through. A rooster's tail feather perhaps?
So, the difference between a sling and a cocktail in that first definition was a dash of bitters through the cock's tail.
Just what we needed--another birth of the cocktail theory.
In case you're wondering, that's Eben Freeman holding one of his beautiful old bitters bottles at Tailor in New York City.