Remington called today's replica "The New Model Army Revolver"; it was introduced in 1863 but carried the 1858 patent stamp. It was an improved version of their "Old Model Army" introduced in 1861. That gun was itself a redesign of the "Remington-Beals Army Model Revolver", first introduced in 1860. This was the first in the line of what we now call the 1858 Remington.
Several years ago Pietta actually made a replica of the Remington-Beals Army Model; it is distinguished by the barrel threads at the breech end being covered by the frame, a characteristic that caused even worse fouling problems than the more modern replicas have. The Pietta version also has safety notches in the cylinder, a feature that was actually not added until the 1861 design.