Your life-long interpretation is correct: à la carte means you order the dishes you want from a list. The “carte” (des mets) is the list of what you may order. However, it should be clear whether you are looking at just a list of items and their prices, or whether you may order an appropriate “menu” or “formule”, which may have a few options like @doreen describes. As for the “menu”, a simple one would be, e.g., one entrée + one main dish + dessert/cheese for a fixed price which may be significantly cheaper than ordering each item separately. If your party wanted an extra salad, or if drinks are not included, then you would order them à la carte. The distinction being made is à la carte ⟷ à prix fixe (even if, strictly speaking, the fixed-price menu had a couple of options)
Translation is sometimes confused by the fact that “menu” in English and menu in French have slightly different meanings.
If you’re dining à la carte, there will be a long list of dishes, each individually priced, from which you make your selection. In English this list is referred to as the menu; in French it’s la carte.
The alternative is table d’hôte; a set meal for a fixed price with relatively few options, or none at all. Le menu is the list of components of the set meal.
In other words, le menu doesn’t carry any connotation of choice; it’s just detailed particulars of the meal that you will be served. Whereas in English “menu” suggests a list of options; the whole point of the menu is to inform you of the choices open to you, and it has come to be used in that sense in a wide range of contexts that have nothing to do with dining.
They can do whatever they want with la carte, as long as the menu promises me a big container of au jus. 