The confusion comes in the color versus the process. I've seen some recipes call for "cook down onions until translucent and caramel in color". This is the most accurate definition of what most recipes want when asking for caramelized onions.

However the true use of the word "Caramelized Onions" are onions that have gone through the cooking process of caramelization. This process requires three things: sugar, heat, and time. Just like making candy caramels, you are performing the same process to onions. This makes true caramelized onions actually sweet, with savory notes throughout.

Most recipes bastardize the word "Caramelize". They tell you to caramelize onions, when in reality you don't actually want caramelized onions (which in many cases would actually ruin the recipe), instead they want softened onions with browned to a light caramel color, which is a completely different thing.

This is like a recipe needing turkey but asking for chicken. Like sure, it will probably work, but it is clearly different and not entirely substitute.

French Onion soup is the most common use of real caramelized onions, which is sweet and savory soup. It could not be made with browned onions, that would ruin it. Coq-a-vin is another popular recipe that uses caramelized onions, which complements the prominent wine flavors of the dish into a sweet, savory, gamey (usually made with game hens), and earthy delight of a meal. The caramelized onions are what pull that dish together and critical part of the recipe. This meal takes a long time to cook, similar to a gourmet version of American potroast.