What troop categories were used historically?

This is obviously not intended to be about every possible order of battle through history, but is asking for a general sense of the standard.

The typical organization of army units is as follows:

Company (Roman century): 50-100
Battalion (Roman cohort): 400-800
Brigade (Roman legion): 1500-3000
Division (Roman army): 8000-10000

This is just a very rough and general idea of the most typical military elements.

An important concept to be aware of is the idea of a permanent versus a changeable unit. In most armed forces there is some key unit that is considered permanent and is often named specially. The other units are considered changeable. For example, among the Romans, the permanent unit was the Legion, but among the colonial British the permanent unit was the Regiment, which was about the size of a brigade. Usually a soldier will identify with his permanent unit. So, for example, Roman soldiers would think of belonging to so-and-so Legion, but a British soldier would describe himself as belonging to Regiment XYZ. The British had regimental ties which they used to wear in the old days, so you could recognize old soldiers by the pattern of their ties.

I would argue that for most of history the reasons for having different "troop types" in an army were socio-economic in origin, with the leadership/royalty/aristocracy forming one broad class of troops (for example chariot-borne royal noble warriors in the ancient near-East, China and Gallic Britain, the hoplite class in Greece, medieval European knights etc) providing the hard core of an army and the masses of serfs/poor acting as some kind of cannon fodder cannon fodder.

So in ancient times battle tactics would be derived to make the best use of the troops that your culture (and your culture's affluence) has given you. This is hugely different from today, where modern militaries develop tactics and then train/equip their soldiers (within the limits of the state's affluence) to implement those tactics.

Warning: The following is a huge oversimplification of a better part of military history.

You are right that the "modern" terminology arose in the early modern period. The reason it came about was that it was precisely during this period that concepts such as "combined arms" came to be more broadly applied, and military science started to form.

This necessitated a terminology to classify different types of troops so that one could come up with a nice theory of how they should cooperate.

In most earlier periods, things like tactics and doctrine were driven by "conventional wisdom" which evolved very gradually with only occasional leaps when a doctrine much better suited to pertinent conditions evolved somewhere.

This "conventional wisdom" would often simplify things to the point where only one arm was considered to be decisive for the result of a battle/campaign, and all the others were seen as accessories. Which arm this was varied historically, but this is the reason why, when reckoning the size of armies, Greeks would mostly just count hoplites (hence the legend of "300 Spartans at Thermopylae"), high mediaeval armies would count armoured knights, et cetera.

So as a general rule, you would have one arm that would be considered your "field force" or some such, with other assorted units that you would think of as "support", "scouts", and so forth, depending on what you actually assigned them to do.

The descriptors for cavalry, foot troops, archers or chariots existed throughought history; it's just that in absence of a formal system of military science, they wouldn't really have been thought of as "categories", or rather, would be descriptive more of the equipment than of the role these troops fulfilled.