Dear CivIIenthusiast!
My knowledge has roots in Shogun: Total War game & Clavell's "Shogun" & branches in some history investigating after that, then...
First of all - Japan become some "united land" only under Imperial rule, that began only in 1867 year. Before that any daymyo (landlord of bunch of provinces) had his own rights over his vassals & own army. It don't help to creat any common terms for any army units at all. "Cultural unity" of Japan began only under shogunate of Tokugawa in ending of XVI century. Until that - the same unit or war-purpose material had different definitions in different part of Japan. But there was some common terms:
In feudal Japan soldier very often was named by his weapon -
Yari & Naginata - for example. Some weapons don't give any names for units, because they are - necessary for distinct class of Warrior: i.e. Katana & Wakisashi for Samurai, or Shuriken & Nunchaka for Ninja.
Your Example is some funny: feudal Japan Army has Yari Ashigaru & Yari Samurai units & all of them was "spear-bearers". You will be surprised, but it isn't common term for "Samurai Cavalry" or "Heavy Cavalry", because cavalry in feudal Japan was scarce & seldom. Most common term in japan chronicles is "Samurai from Shinano", what means "cavalry" as whole.
This definition appeared in Takeda clan - only warrior clan who had some significant horsebreeding territories (and significant Cavalry as result) in Shinano province, but in southern lands (Shimazu or Mori clan lands - in Kyusyu island) this term was completely unknown.
There isn't any common term of Archer in feudal Japan. Reason of it is - any Samurai must be proficient in Archery, because it is necessary condition of Samurai by "Bushi-do" - "Way of Samurai".
In real life the same Samurai can be Yari Samurai, Samurai Archer, Naginata (heavy sword), or Samurai Cavalry. Distinction appeared after decision of local landlord who keeps concrete Samurai fiefs. Then - all Samurai from Aki province was counted as Naginata from beginning of XVI century, but all Samurai from Omi as - Archers. In reality - every Samurai must have: Katana (long sword), Wakisashi (short sword), Bow with Arrows & Samurai Honour. No one must have Yari (spear), or Horse. If Samurai get Yari (for any purpose) he is demoted to Yari Samurai, because spear was "peasant weapon" for "Bushi-do". Horse Riding skill was mandatory, but not necessary till beginning of XIX century, then this term uncommon as well.
This confusiing situation is main headache for any history-writer describing feudal Japan. In some sense feudal Japan has only one military unit - Samurai, who was all-purposed unit. Only in shogunate of "Ashikaga" (i.e. in XV-XVI centuries) some peasants was taken in japanese army as Yari Ashigaru & Ashigaru Musketeers (yes, it is - this term is pronounced as "Mu-shi-ki-ta" indeed!). New Shogun - Tokugawa banned this practice.
Well, you can use "Ashigaru" term only in very short period of japanese history - clans of Minamoto or Taira ("Epoch of Fight between East & West") or "wars of Fujiwara times" - haven't got this term yet, & Tokugawa hasn't got this term already

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Beginning of Fujiwara rule is beginning of feudal Japan history. Ruling of Tokugawa house is end of it. (1867 - to be exact).
I think - most correct thing is: give to units some common names - as "Samurai Archers", or "Archer Cavalry" & forget about autencity. In other case you will have some very weird scenario with some very weird names for units & history elements as some scenary of one very prolific author with German name about Russia's history. I think - if your scenary will be enough interesting for any player (japanese or not) it will have historical autheticity & "national colouring" enough.
I see & realize some historical nonsense in some Eklund's scenarios, but they was done so perfect & player-consuming that in some sense - I can't see them! It is good & marvelous work indeed. If gameplay "consumes you" completely, you just feel some "national & historical flavour" & it is great thing. If you begin to be bored you must find most ugly mistakes in authors design & begin laughing - unmercifully. That is that.
Good luck & any success in your working, Alex.