I also appreciated the proper definition to Men-at-arms. On another note, was Men-under-arms the proper term for a run of the mill soldier?
As a separate add-on, since Axemen/Swordsman are probably the last of the regulars/mass war-fare that their name could be based on a number as opposed to a weapon type? (assuming Champions are individual men, or smaller more elite units)
In this case, I think the soldiers of the empire could be referred to as Platoons. So a platoon is the unit name, and a given platoon will wield sword, spear, or axe, whatever the stylings of their civ will dictate.
This however, is a tricky concept because 1) the numbers of each unit are not defined, and 2) someone could easily wish to consider their archers, horsemen, and mages to be platoons as well (although I honestly prefer "wings" to refer to horse-crews)
It is true that many times a unit is named after the weapons they wield, although often (and in this case it may be preferable) it is wise to name the unit after "how" they fight, or something specific to that unit.
Currently I have always found the term warrior to be misleading, as many cases the name warrior is used in heroic context, while other times it is a general term, although it is never a grunt unit's term, unless its Civ (by the meaning of being the lowliest of units)
There fore, I would propose removing the term "warrior" entirely, and replacing it with grunt. It is hard to say how the switch would be accepted, as warrior being a lowly weakling is probably ingrained into every civver, although it would remove the potential inter-changability of fighter/warrior.
This proposal does keep the previous fighter suggestion, so that it would be Grunt -> Fighter -> Champion.
This is simply my opinion of how a merge should be done if a merger is decided upon.
Personally I quite prefer the current axeman/swordsman, with the Caveat that you add a spearman UU for the Elohim.