So, I thought I should clarify and codify how I'm using military terminology in the City,

First and foremost, I'm borrowing heavily from real militaries. First and foremost, ranks and titles are almost all a direct rip from real militaries. As decisions about world building go, it's often a better idea to borrow something a reader already understands, than have one more new thing to explain. Most people understand the basic relationships between the ranks. Private is the lowest rank, sergeants are more important, lieutenants are incompetent leaders who, for some unexplained reason boss the more competent sergeants around. Captains are your reasonable authority figures, unless they're dashing ship captains. Colonels drink port in fortified bunkers far away from actual fighting. And so on.

This makes the ranks and titles accessible for the reader, and saves their attention to other, more interesting parts of your world.

I'll post another section about the different kinds of army units in the City. But here, I'll explain the ranks.

From lowest to highest rank:

Auxiliary, Enlistee
The lowest rank in the official chain of command, auxiliaries are specialists borrowed from outside the military for specific purposes, and enlistees are civilians recruited 'in the field'. They do not give orders to anyone.

Private
The first rank of a formally enlisted soldier. Only a soldier who has chosen to enlist is allowed to wear the uniform. Note, chosen to enlist. There is a small rank distinction between recruits and graduates of basic training, but this distinction is erased during any active combat situation.
Privates are trained in basic communications, the most basic operation of artillery, and a functional but rudimentary use of both Salamanders and swords.

Specialist
Considered one rank above a private, specialists receive their rank for demonstrating advanced proficiency in a specialized field useful to the military. There are only six fields that this rank will be granted for.

Artillery
Communications
Engineering
Explosives
Medicine
Mechanics

Corporal
Leader of a group, the smallest military unit. Rising to this rank requires some amount of time in the service, usually two years. A corporal is the first real taste of command for most soldiers.

Master Specialist
Held above a corporal in rank, Master Specialists are, as the name suggests, masters in a specialty deemed critical to the Military. Among other talents, a Master Specialist is considered qualified enough to teach and certify someone to the rank of Specialist. Master specialists are distinctly uncommon, as anyone earning the rank is usually considered command material and is often trained up to sergeant. Officially and functionally ranks above a corporal.

Sergeant
The highest ranking non-commissioned officer. Sergeants often have specialties, such as Instructor Sergeant, a specialist ranking, logistics, etc... in that case, a sergeant is addressed with the additional title only if their work currently pertains to the specialty. A sergeant is only an instructor sergeant while she is teaching, an artillery sergeant is a gunnery sergeant only while he is directing a Valkyrie, etc. Master Sergeant is a title given only to a sergeant who has a master specialist designation, and like a specialist sergeant, the title is only used if the specialty applies.

During moments when a special title applies, the specialist sergeant is considered to rank above a sergeant, but beneath a commissioned officer.

The commission: Above master sergeant, rank requires the symbolic surrendering of a commission to the City. Usually just a single lottery token, though often a prized childhood toy. In exchange, the officer is granted a commission, and granted the formidable authority the rank represents.

Lieutenant
I'm a little hazy on the entomology of the word, but I believe it derives from two words that eventually became left and tenant, and the rank literally meant that a lieutenant was responsible enough to be left to tend a group. The lowest and most inexperienced officer, lieutenants are granted responsibility over a platoon, which is made up of two to four squads.
The group of a platoon is usually considered the largest group of people a single officer can be expected to get to know well. Lieutenants do not directly oversee issues like billeting and training, but are expected to ensure that they are done, and done well.

Among other privileges, only a commissioned officer is allowed to command a Crafter.

Captain
The highest army rank expected to be visible on the front lines, a captain commands a company, made up of three to six platoons, usually 120 people. The company is considered the largest group of people a single officer can be expected to know by name. Captains are also the lowest rank that can be expected to take overall command a battle space, for instance a group of Valkyries firing at a Golem should be commanded by a captain.

Major
Commander of a batallion, usually five to eight companies. A major is not expected to know the individual soldiers under her command, but can be expected to know the ranks by name all the way down to master specialist, for the sake of knowing the strengths of each company. Majors usually direct battles from a safer distance, in order to coordinate between several companies at once. Majors usually have a platoon attached directly as support staff, usually over-strength and comprised heavily of medical and communication specialists.

Colonel
Commander of an army, consisting of five to eight battalions, usually about 3600 soldiers.

Secretary to the Lord Captain
Originally a joke title. During the First, as the captain of the volunteer group the Cadavalan Rangers was saddled with responsibility for the defence of the entire City, Ivalia Dustal, a secretary from the Bureau of Lands and Agriculture became Captain Cadavalan's principal secretary and head of logistics. When the captain had to go to a specific front and needed the other side of the City's defence overseen, he appointed his secretary to command it. Unsure of how to address her, Secretary to the Lord Captain became an unofficial rank, and when Lord Captain was made permanent, so was Secretary.

Unique in the army, a Secretary to the Lord Captain is not a permanent rank. A secretary derives his authority from the direct appointment from the Lord Captain, and the rank only survives as long as the current Lord Captain.

A Secretary is usually in command of a battle space, rather than a specific number of groups. There are usually four, one for each compass point, and they oversee the defence of the City under the directives of the Lord Captain.

Lord Captain
Originally a joke title. During the First Invasion, a company of volunteer watchmen and scouts calling themselves the Cadavalan Rangers were lead by the informal command of Andre Cadavalan. The company proved to be a source of strength and resolve in an invasion no one was expecting, and the company was set up in progressively higher points of authority because people listened. A bit of snark from a field enlistee saw Captain Andre addressee as 'My Lord', cuffed on the back of the head by a companion, and then addressed as 'Captain'. Overheard by half of Andre's company, the name stuck. By the end of the First, nearly everyone who fought referred to him as the Lord Captain, and when Parliament decided to make the army a standing institution, the rank stuck and survived Andre's passing.

The Lord Captain is the supreme commander of the armies of the City, and effectively a dictator during an invasion.

In the above doodle, you can see the insignia for each rank and specialty. For Non-commissioned officers, their ranks are worn on badges sewn on the shoulders of their coats. Commissioned officers wear their rank on the pommel of their Coldstone sword.

Quick note: for my own convenience, I showed what a communication specialist designation looks like along with regular ranks. If the soldier were an artillery specialist, the dot dot dash logo would be replaced with a canon.