Here there are people arguing about historic names of mercenaries companies, while in the game there is an actual battle scene about soldiers pushing each others with shields in a deadly battle as a means of wining...Pal, you are making a meta-history here over narrative... We can't make a just comparative between two man in which their circumstances were a hell different... Rowan lives his reality in what we can identify as what we called in our history as "The Ancient Regime" meanwhile Napolion lived near half of his live during "The New Regime" (and participated in the demise of the Ancient Regime). Taking note that, Napolion came from a low nobility family in a decadent monarchy with more access for education than most ppl in Rowan's circumstances, who seems to live a few centuries before the period Napolion lived, the game stated the importance of the food production, something that was more of vital importance in economic terms during Middle Ages, so, we can assume Rowan is living in a period like this, moreover, there are hints about the existence of the Three States full of shit of that period. Respect to Rowan actions, it was precipitated, non calculated, imprudent to his very own reputation, he had some nobles on his side with high esteem for him, but he never could acknowledge them nor consider them.
And most probably it is because of that novel that was popularize like that, the company you are referring was originally called Great Company of English and Germans, to later English Company. In Italy, this company was known as White Company of the Falcon and White Company to short, named as most of it's members had symbols and heraldic with that color. Moreover, this company was founded several times and didn't had the same leader each time. And more probably Arthur Connan Doyle take the name from the italian source (as John Hackwood came to be it's leader during the period the company came up to be founded again in Italy), and as you stated, the mercenary armies often are called by many names, one of them are White Companies because this was not only the most famous, but because historically was founded in different places around Europe, other names for them are "Great Companies" because often many of them take that title, other way to refer to them are Free Companies with Free to be understood as "Free" because non of them formed alone by it's own, they always served to some kingdom or noble, in other words, some kind of political power.
IT HURTS!!! PLS GUYS GO MAKE A GAME!!! I NEED SOME GOOD MEDIEVAL RPG!!!