Lots of medieval and ancient armies maintained a distinction between 'light' and 'heavy' troops, but 'why' they were 'light' or 'heavy' could vary. At the same time, I would argue it would be a mistake to try impose our contemporary perceptions of troop structure and divisions onto the past.
If we look at the following translation of the Strategikon of Maurice (6th century AD):
We find that the Byzantine military establishment of the period understood that soldiers with different types of equipment had different roles to play on the battlefield, and so had different designations:
"2. Training of the Individual Heavy-Armed Infantryman
They should be trained in single combat against each other, armed with shield and staff, also in throwing the short javelin and the lead- pointed dart a long distance.
They should be trained in rapid shooting with a bow, using a lance set up a good distance away as a target. They can shoot in either the Roman or the Persian manner. They should be trained in shooting rapidly while carrying a shield, in throwing the small javelin a long distance, in using the sling, and in jumping and running."
In this case, the division between 'light-armed' and 'heavy-armed' was based on whether an individual was intended to fight in close combat or at range, with no requirement that one should wear armor. Even then, the 'heavy' troops maintained some capability to fight at a distance. Sometimes, such a sharp distinction was not present amongst other peoples. For example, in the early part of their history, the Achaemenid Persians used troops that could function in either role. According this translation of Herodotus:
https://www.gutenberg.org/files/2456/2456-h/2456-h.htm
Persian infantry were equipped in the following way:
"Now those who served were as follows:—The Persians with this equipment:—about their heads they had soft felt caps called tiaras, and about their body tunics of various colours with sleeves, presenting the appearance of iron scales like those of a fish, and about the legs trousers; and instead of the ordinary shields they had shields of wicker-work, under which hung quivers; and they had short spears and large bows and arrows of reed, and moreover daggers hanging by the right thigh from the girdle: and they acknowledged as their commander Otanes the father of Amestris the wife of Xerxes."
It also important to note that such warriors wore armor, and so would be considered 'heavy' infantry if we used our modern system of classification, but they would often fight primarily as ranged troops, and so would be 'light-armed' by 6th century Byzantine standards. The Persians would set up shields as a wall, and then bombard opponents from a distance. If they enemy came close, they would fight with spears and swords, and so then would be considered 'heavy-armed':
"Thus the Lacedemonians and Tegeans were left alone, being in number, together with light-armed men, the former fifty thousand and the Tegeans three thousand; for these were not parted at all from the Lacedemonians: and they began to offer sacrifice, meaning to engage battle with Mardonios and the force which had come against them. Then since their offerings did not prove favourable, and many of them were being slain during this time and many more wounded,—for the Persians had made a palisade of their wicker-work shields and were discharging their arrows in great multitude and without sparing,—Pausanias, seeing that the Spartans were hard pressed and that the offerings did not prove favourable, fixed his gaze upon the temple of Hera of the Plataians and called upon the goddess to help, praying that they might by no means be cheated of their hope, and while he was yet calling upon her thus, the Tegeans started forward before them and advanced against the Barbarians, and forthwith after the prayer of Pausanias the offerings proved favourable for the Lacedemonians as they sacrificed. So when this at length came to pass, then they also advanced against the Persians; and the Persians put away their bows and came against them. Then first there was fighting about the wicker-work shields, and when these had been overturned, after that the fighting was fierce by the side of the temple of Demeter, and so continued for a long time, until at last they came to justling; for the Barbarians would take hold of the spears and break them off."
In terms of looking at the military practices of the past, one should not try to evaluate or categorize them according to our standards, but rather abandon all such existing understandings and try to work out what the people within that culture thought about warfare: what did they believe was the 'ideal' or 'practical' way to fight, if there were divisions based on certain weapon types or battlefield roles, did they communicate that this was a deliberate choice on their part, or was it something that naturally occurred because of economic of social factors and so they did not give much thought to it?