Granted.True. I added them to my master list. An interesting point that translates to a different label depending on the personal preferences of the fighter.
Man-at-arms is a term i had forgotten about. Bodyguard is one that i somehow managed to forget to add to the list as well.
As mythical as the lionized puritanical samurai or the virtuous knights in shining armor.
The firearm instructor is an interesting example, since military training is not at all required, but a bonus. Their job would be teacher, what they teach would be a secondary detail. Though any actual teacher would tell you what they teach matters a great deal, in game terms it's just fluff.
Anyone who takes pride in their work trains to get better at their job, it doesn't matter who or what they are. And if you're good at anything you should always get paid for it. And the better you are, hopefully the better you get paid.

The key point i'm trying to refine here is the least common denominators here. this means working out my own personal balance of real world practices with gaming trends.
So right now we're talking about people who fight FOR money, not just people who know how to fight. So let's look at what separates them, ignoring the names or ideosyncricies. And these have to be jobs, that means getting paid.

Fight for themselves - Any fighter with a notion for ideals will say that they fight for anything other than money. but they can't eat pride. So they find other jobs or live off the grace of donations like a religious group might. However they may just as easily be bandits, fighting to get stronger and to raid others of their wealth and property.

FIght for other people - A fighter who fights when others need them to and prefers to live in peace not fighting in the mean time defines many heroic figures. Guardsmen, militia, vigilantes, retired soldiers. Their jobs are what they do in the down time, fighting is their duty which may still get them paid or compensated by a community. How often they fight does not matter.

Fight for the state - A warrior who fights for duty and believes that the kingdom, the government or whatever ruling body is the best place to learn, the most profitable, the most reputable or the most dignified. Fighting for the state gives the warrior authority and permissions that other fighters may not get or be thoroughly denied. Soldiers, police, knights, samurai, ninja, royal guard. What separates these fighters from others is not their prowess but rather their political connections and access to resources. A knight, samurai, marine once removed from the contexts of their time and place can be argued as identical in motivation.

Fight for money alone - Mercenaries, assassins, gladiators, duelists and others who find insufficient motivation in honor or morality and focus entirely on greed or survival are likely to spend most of their time fighting. they may fight for anyone without gaining the advantage of working with those groups.

In these terms the things that separate fighting as a job is separated by association and motivation. And in game terms those are fluffy terms. Even what weapons they use are not relevant to the separation in categories, making it personal preference. And by that logic I shouldn't have archer or pikeman on my final list.
In Monster Hunter, characters only have 2 classes, depending on whether they use melee or ranged weapons. In many other RPGs i can think of there is no class or job, just a collection of skills and equipment, which may be the way i'll end up taking it. Most strategy RPGs separate classes/units based on the weapons they use or the type of spells or skills they use. Which also means that if I do the same, I can just pick and choose from games like Final Fantasy Tacitcs, Disgaea, XCOM, Warcraft, etc etc and cherry pick the ones i like until i find a list i think is balanced. And whilee that is cut and dry easy to do, i find it to be unnecessarily limiting and against the goals of my project.

AND let's not forget the good old job description of "adventurer" which effectively means anything and everything that involves going somewhere else and coming back alive with material or information. Bards, wizards, clerics, barbarians can all be adventurers. An since an adventurer with no combat abilities might be called an explorer, archaeologist, scout, or a curious child and don't pay anything unless someone comes back with something useful, calling it a 'job' can be called dubious too.

If this 'Job List' Stands as nothing more than an exhaustive list of sticky star labels a player can choose to apply to their character to not, that's just fine with me. Turning them into a concrete list of a combination of stats and in game effects forces the math into a shape unnecessarily rigid.
HOWEVER that does not mean i won't end up doing it anyway ^^
This discussion here is about building up the exhaustive list, listening to differing options (like yours, which i am grateful for) and accounting for those (even if i do not agree with or accept them) until i can find a least common denominator that will make it easier for a player to create ANY of these labels, as they subjectively imagine them.