This gun has some contradictions.
There were 3 'large frame' Remington revolvers made before and during the Civil War:
1) the 1860 Remington-Beals Army Model Revolver, about 1,900 made in 1861.
2) the 1861 Remington Old Model Army Revolver, about 9,000 made in 1862 & 1863,
3) the 1863 Remington New Model Army Revolver, about 122,000 made in 1863 through 1875.
1. The serial number indicates it is an 1861 Remington Old Model Army, the second model of the 3. Further, from the number it would be an early manufacture of that model.
2. The stamp on the barrel also indicates it is an 1861 Remington Old Model Army. This is the only model that stamp appeared on.
3. The frame appears, from the photo, to be from the 1860 Remington-Beals Army Model, which was the first of the 3. The frame covers the barrel threads at the breech end. Correction of this defect was the main reason for changing to the Old Model Army. It's possible that, being an early manufacture Old Model, it was built with remaining frame stock from the earlier Beals design. However, I've never seen reference to such a transition model. This contradicts 1. and 2. above.
4. The loading lever does not appear (from the photo) to have the relief allowing the cylinder pin to be pulled out with the lever stowed. This was a feature of the 1861 Old Model Army design only. This contradicts 1. and 2. and supports 3. above.
5. The hammer notches on the cylinder were indeed introduced late in the manufacture of the 1861 Old Model and appeared on all the 1863 New Model guns. It's possible, in fact likely, the cylinder is not the one that came with the gun originally. This contradicts 1. and 3. above.
It must be remembered that armories mixed and matched parts in repairing guns throughout the war. I don't find it unusual that this gun appears to have features that appeared on different models. We tend to want to believe that every antique is as originally manufactured, an assumption that is far too often an unwarranted simplification.
I have one of the Euroarms replicas of the 1860 Remington-Beals Army Model Revolvers. It is a contradiction in itself, as it has the cylinder hammer notches of the later 1861 and 1863 models and is stamped 'New Model Army' on the barrel. The OP's gun does not match the Euroarms replica as regards the barrel annotation.
(Please excuse the erroneous date in the photo caption)