I thought early west european medieval army composition outside of scandinavia was something like this;

the local lord with his vassal knights and their men at arms, these are pretty much indistinguishable from each other apart from the amount of cash expended on armor. they all carry as many sharp or blunt objects as possible.

the peasant levy, no armor, no formal training, and as many sharp or blunt objects as they possibly can afford and carry. the only kind of specialisation would be if anybody brought bows.

no other troops, be they professional or part-timers. no units made up of specific weapons.

then of course later, you get some at least semi-professional troops of crossbowmen, longbowmen and eventually pike formations which are, in fact, distinguished by carrying a special form of armament and being grouped according to weaponry, but until that, youre either a man at arms(including the nobles) or a peasant or townsman, and you carry whatever nasty spiky piece of metal you can get your hands on.