Fetus, each drink has its own recipe and there really aren’t that many options. But there are thousand of drinks and variations of drinks.
Some of the most common options I remember from my bartending days:
“Double.” Double the booze, but not double the mixer. A stronger drink.
“House.” Just the regular stuff they serve by the glass. Whatever they usually serve, you don’t care. Usually used with wine.
“Mist.” Booze served over crushed ice.
“Neat.” Booze served plain, un-chilled, no ice.
“On the rocks.” Over ice.
“Splash.” A small squirt of something. Whiskey, neat, with a coke splash."
“Straight up” or “up.” This one can be tricky with finicky customers. It means “neat” that is, plain booze in a glass, but the glass could be chilled and/or the booze chilled first. It’s best to tell your bartender what you prefer. “Whiskey, straight” could get you a chilled glass, or whiskey from the freezer, or whiskey shaken over ice and strained into your glass. Customers love to bitch about this one.
“Virgin.” No booze. Don’t be a smart-ass and ask for “vodka, neat, virgin.”
“Water back” or “back.” A glass of water or something else served with your drink. Basically the same as a “chaser.” “Scotch, neat, with a water back, please.”
“Well.” The everyday stuff the bartender pours from the rack right in front of him. “Whiskey on the rocks, well is fine.”
“With a _____ Chaser” A small drink to accompany your main drink. “I’ll have a Bloody Mary with a beer chaser, please.” You should get a small house beer along with your Bloody Mary. “Scotch, neat, with a Sprite chaser, please.”
Was that what you’re looking for? If you google up “bartending terminology” you can find lots more.