Howdy
I cannot provide all the answers, but perhaps I can add a little bit of insight.
The New Service was made for a long time, from 1898 through 1944 and it was chambered for no less than 11 different cartridges.. You can do the math on how many years that was. I would not be surprised if during that time period there was more than one scheme for marking the caliber (cartridge) on them.
This pair are quite early, the 45 Colt New Service at the top of the photo was made in 1906 and the 44-40 at the bottom left the factory a year later in 1907. Notice how the trigger guards do not blend smoothly into the frame but there is a hard line where the trigger guards meet the frame. Later models, as well as the 1917 military model blended the trigger guards into the frame.
Sorry that this photo is a little bit blurry. This one is clearly marked for 45 Colt. That is what it is chambered for, a 45ACP slides right through the chambers. Also, 45 Colt is a rimmed cartridge. The cartridge headspaces on the rim. 45ACP headspaces on the case mouth. Model 1917 revolvers (chambered for 45ACP) have more space between the rear of the cylinder and the frame, around .090, to allow the use of moon clips. This revolver has about .075 between the rear of the cylinder and the frame, so the 45 Colt rim will seat properly.
This is the 44-40 New Service. It is simply marked CAL .44. As I said earlier, it left the factory in 1907. The 44 Special cartridge did not appear until about a year later, when Smith and Wesson created it for the Triple Lock.
Here is the proof in the pudding of what cartridge this old girl is chambered for. Notice a 44-40 cartridge chambers completely, a 44 Special does not. (Ignore the black stripe on the 44-40, it is one of my Black Powder rounds.)
44-40 is wider at the rear than 44 Special. About .462 for the 44-40 vs about .452 for the 44 Special. You can see how there is a bit of slop around the base of the 44 Special cartridge in this photo. But 44-40 is a bottleneck cartridge, 44 Special is not. Up at the case mouth the 44 Special is still .452 in diameter. But the 44-40 is only about .440 in diameter at the case mouth. The 44 Special, with its wider case mouth will not chamber all the way in the tapered 44-40 chamber.
Hope this helps a bit. But it probably makes things more confusing. Like I said, I would not be surprised if there was more than one marking scheme for the New Service over its long production life.
By the way, the New Service and the Model 1917, which was built on the same frame, were really big guns. I don't own a 1917 Colt, but look how much bigger this New Service is than a S&W Triple Lock. The frame is huge, it is the biggest revolver I own.
Here is the same New Service and a S&W Model 1917, for comparison. The S&W Model 1917 was built on the same frame as the Triple Lock.