All of them roughly map onto the more modern category we typically use for anything involving mixing spirits with something acidic, sweet or bitter: cocktail. "Sling" still survives via an old-timey British cocktail, the Singapore Sling. Says Wikipedia: "a sling was originally a North American drink composed of spirit and water, sweetened and flavored."

As for pepper in rum, I had to look that up too. I suspect that Benjamin Rush would've been referring to something similar to this, because Jamaican rum was so dominant in the British colonies: https://www.sbs.com.au/food/recipes/pepper-rum

Edit: Confusingly, this source from the 1780s [1] says "rum, whisky, brandy, gin, stinkibus, bitters, toddy, grog, slings and fifty other liquors, all come under the denomination of spirits." But this is presumably what Rush had in mind because it's direct from his book: "After a while nothing would satisfy him but slings made of equal parts of rum and water, with a little sugar." [2]

[1] https://books.google.com/books?id=VyRTvziJuecC&dq=sling%20sp... [2] https://books.google.com/books?id=PnZVb3esMk4C&dq=sling%20sp...