The original 1858 Remington .44 revolver model was called the "Remington-Beals". This was the model which was in production at the start of the War Between the States. In 1861, the loading lever was slotted to allow the cylinder base pin to be withdrawn without the loading lever being lowered and a new "1861" patent date stamped on the barrel marking. This is sometimes refered to as the "1861 Army". It became known as the "Old Model Army" in 1863 when Remington made some changes, chiefly removing the slot in the lever, and renamed it the "New Model". The barrels on these were marked with the original 1858 patent and the words "New Model".
General specs and configuration were pretty much identical for all of them, otherwise. I'm not sure where, when, or how many of which were produced with the gain-twist and/or "Progressive" rifling which was touted as producing much-enhanced accuracy with both RBs and conicals. "Safety" notches were added to the cylinders along the way somewhere, too, but not sure of exactly when there either. When we're talking about replicas, it's virtually a moot point anyway. With the exception of only a couple of high-end limited production models, none of the current replicas offer the "exotic" rifling, and all of them have the notches.
IMO, the nomenclature changed when the basic specs were altered as technical advances in both ballistic and manufacturing arts were incorporated into the design, as well as for pure marketing hyperbole. Declaring a "New Model" was an even bigger marketing tool then as it is today.