Delicious variations abound, so it is hard to make absolute statements of what is "officially" a ladyfinger. In my pastry class, we were taught that the classic recipe for ladyfingers is close, but not identical, to genoise. Store-bought ladyfingers can be anything from, essentially, a meringue to a shortbread cookie.
Foam-based batters -- i.e., those that do not use chemical leveners or yeast -- can be divided into two categories: whole-egg and separated-egg. The main difference in results is that the latter tend to be stiffer and can better tolerate being overcooked a little. The primary example of whole-egg batter is the genoise cake. Because the whites are whipped together with the yolks (and sometimes extra yolks are added), it is a finicky batter, requiring more skill to assemble properly than others, but when it works, it is more tender. Separated-egg batters are far more common, as they are easier to assemble; the key is that you whip the whites into medium-to-stiff peaks, then fold it back into the rest of the batter. Examples include chiffon, dacquoise, and the French "biscuit" cakes.
Most recipes for ladyfingers use a separated-egg batter; others use a simple meringue. The meringue version will make a crisp cookie, the version with with the yolk, soft. Either will work well for uses like tiramisu or trifle.
Here is a recipe for ladyfingers from Allrecipes.com and a recipe for genoise from Epicurious. For my ladyfingers, I add a pinch of salt and some flavorings, such as vanilla or nut extract, but the base recipe is essentially the same.