One of my daughters married a Brit and lives in the UK. Lots of things are considered part of some "kit". You get used to it if you know the source.
My pet peeve is calling cartridges calibers.
I have several guns in 45 caliber but they are different cartridges. I have 45 Colt, 45 ACP, and 460 S&W Magnum guns. All 45 calibers yet different
That doesn’t make it sensible. Method of operation makes a lot more sense if you are talking about the procedures. Operating manual makes more sense if you are talking about where it is all written down. Using the same term for both is silly.
There are many things that are called by more than one term. Cylinder stop and bolt in revolvers is one that comes to mind. This isn't unique to firearms either.
I spent over 40 years in the Army (AD, Guard, and Reserves) and over 30 years in law enforcement and more often hears the term manual of arms applied to a weapon's operating instructions.
The operator's manual is where you find the steps in operating the weapon aka the "manual of arms". All the "operating instructions" for a firearm are lumped into the term "manual of arms".
I was under the impression that manual of arms was those commands of when and where to manipulate our rifles in drill and ceremony in the military. As I remember, there were only about 5 or 6 commands, so it was pretty easy for most of us. For those having difficulty. extra attention was given. "WHAT SIDE WAS THAT, PYLE"!!! Remember that? That's what I remember as Manual of Arms. Publications about our weapons were called field manuals and technical manuals. Or maybe I'm not remembering correctly. I do remember my right from my left, so that's good.
You are correct that manual of arms is the term applied to how you carry and handle the rifle in relation to Drill and Ceremonies outlined in FM 22-5.
Manual of arms is also used to describe how to load and fire the rifle. Google Von Steuben's manual where he lists the steps to load and fire. Von Steuben didn't invent the term. You can find earlier references.
It may be an old term but is still in use.