British · Kahlua
Kahlua is the coffee liqueur nearly every espresso martini recipe means when it says coffee liqueur. It is made in Veracruz, Mexico from arabica coffee and rum, and it has been the default since the drink was invented in London in the eighties.

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The one thing that matters
One ounce is the number. Two makes the drink sweet and heavy, and the espresso stops coming through. If you want it sweeter, add a quarter ounce of simple syrup instead.
Yes, and it is the standard recipe. Two ounces of vodka, one ounce of Kahlua, one ounce of hot fresh espresso, shaken hard with plenty of ice. Kahlua is what most recipes mean when they say coffee liqueur.
One ounce to two ounces of vodka. Any more and the drink turns sweet and the coffee disappears. If you want it sweeter, add simple syrup rather than more liqueur.
No, any coffee liqueur works. Kahlua is the sweetest and most widely used. Patron XO Cafe is drier and more tequila-forward, which makes a sharper drink with less sugar.