All stories about New Orleans drinking lore must now begin with the same disclaimer: Cocktails were invented there. Except they weren’t.

For years, many considered the city the birthplace of the cocktail. The tale is immortalized by Stanley Clisby Arthur in his lively 1937 book, “Famous New Orleans Drinks & How to Mix ’Em”: In the first half of the 19th century, Antoine Amédée Peychaud began to mix his spicy, neon-red bitters — which still bear his name — with spirits. “Peychaud had a unique way of serving his spiced drink of brandy,” Arthur wrote: He poured it into a coquetier, a double-ended egg cup. Coquetier (pronounced ko-kay-TAY) became “cock-tay,” then cocktail. “Presently, all New Orleans was drinking brandy-cocktails,” Arthur wrote. “In such fashion did the inconspicuous little crockery coquetier or egg-cup become the christening font of the cocktail.”

It’s a fun story — but it’s not the real origin of the cocktail. (The indefatigable historian David Wondrich has traced the word a few decades further back, probably to upstate New York, or possibly England, depending on how you define a cocktail.)

No matter. If New Orleans wasn’t the birthplace of the cocktail, it is certainly its spiritual center — where it is most passionately celebrated and where some of its most wondrous and varied iterations were invented: the Sazerac, the Ramos gin fizz, the Vieux Carré. “The quality of mixed drinks as served in New Orleans has always appealed to the sophisticated taste,” Arthur wrote, “but the drinks and their histories are forever linked with the past of this pleasure-loving city.”

Over a few winter evenings in New Orleans, I sought out some of those classic versions at decades-old bars and more recent additions, where bartenders are reinvigorating the cocktail scene amid a broader post-Katrina revival.

The links to the past that Arthur described are evident everywhere. At Compère Lapin, a restaurant and bar in the Warehouse District and the newest spot I visited on my crawl, I chatted with the charming and gregarious bartender Abigail Gullo. She made me, upon request, a De La Louisiane, one of the city’s lesser-known inventions — a rye-based drink punched up with Peychaud’s bitters and absinthe.

I asked Ms. Gullo why Compère Lapin doesn’t have classic drinks on its menu. “Because we can make them,” she said, and everyone knows it. “And then if they have a Sazerac, what do they have next? I can guide them.”