The Rum & Coke is the drink nobody needs a recipe for, which is exactly why most of them are bad. Too little ice, too much cola, a flat soda gun, and a lime wedge that never touches the drink. Get the ratio and the lime right and it's a genuinely good highball. The lime is also the whole difference between this and a Cuba Libre.
A Cuba Libre is a Rum & Coke where somebody bothered to squeeze the lime.
History
The standard origin story puts the first Cuba Libre in Havana around 1900, mixed for American soldiers after the Spanish-American War. The main evidence for it is a sworn statement from a man named Fausto Rodríguez that Bacardi ran in its own advertising in the 1960s. Bacardi sold the rum. Weigh it accordingly.
What's beyond argument is that the combination went mainstream in the US in the 1940s. The Andrews Sisters' 1945 hit "Rum and Coca-Cola" was built on a Trinidadian calypso and ended up in a copyright suit, and by then the drink was a standard bar call. The lime-free version is simply what that order turned into at a busy bar.
The Spec
Two ounces of rum, four ounces of cola, a glass full of ice, a lime wedge. Squeeze the wedge into the drink before you drop it in. I'd call that the Rum & Coke done right, and plenty of bartenders would call it a Cuba Libre. They're not wrong.
Which Rum
A light or gold Spanish-style rum is the classic pick: Bacardi, Havana Club, Flor de Caña 4. An aged Jamaican rum like Appleton Signature gives you more funk and stands up to the cola's sugar. Spiced rum turns it into a different drink, sweeter and vanilla-heavy. Fine if that's what you want. Just know it.
Cola and the Lime
Cola made with cane sugar (Mexican Coke, in the glass bottle) tastes cleaner and less syrupy than the fountain stuff. Whatever you use, it should be cold and freshly opened. The lime does real work. Its acid cuts the cola's sweetness, and without it the drink tastes like flat dessert by the second sip.
Bottom Line
Fill the glass with ice. Measure the rum. Squeeze the lime. That's three habits, and they're the difference between a drink and a punchline.