As mentioned in a couple of comments, this almost certainly means egg white, which is used in some shaken drinks to create a stable foam on top of the finished cocktail. It can also "round out" the flavor of a drink and prevent them from seeming overly sweet or tart. Neither of these is explicitly identified as a shaken cocktail, but that's the usual technique for drinks that include both sugar and citrus, as both of these do.
Some recipes use as much as an entire egg white, but much smaller quantities can be used if you crack and separate eggs before service, storing the whites chilled in a squeeze bottle, or mixed directly into a syrup you're making in bulk for one specific drink, a technique sometimes used to speed up service in high-volume bars. (For the "English Writer", you'd combine and keep chilled the non-alcoholic ingredients: grapefruit, lime, honey, and egg white. This allows your bartender to measure only 3 ingredients when making the drink rather than 6, which speeds the process and increases consistency.)
The abbreviation to simply "white" could mean any number of things: it could be they're using an egg substitute instead, or don't want to raise health concerns about the egg (which are minimal, but guests occasionally get squeamish) or the recipe simply got cut off (bartenders tend to list recipes from largest quantity to smallest). Without the exact context, it's difficult to say why the "egg" was left off here, but it wouldn't surprise me to see egg along with the other ingredients in both of these drink recipes. It's commonly used in cocktails and I can't imagine what else they might mean.