As I figured, the gun does not have the "44 Winchester" marking on the left side of barrel. Top breaks in 38 are more common, but out of 32, 38 and 44, the most common 6in would probably be a 44. For the smaller cals, 3.25 in and 4in barrels were more common. Since the cylinders on the 44 DAs were elongated for the 44-40 (which came out later than the 44 Russian), just measure the cylinder. If it is 1 9/16 in long, then its a 44-40, and if its a 1 7/16 in cylinder, then its a 44 Russian. The serial number you gave was within both models, with the 44-40 being 1-15k or so and the 44 DA Russian being 1 to approx 53k.
There isn't any finish left, and the grips are not originals. If the gun works, it certainly has shooter / CAS value. Its of course worth more as a 44-40. The 6in barrel in these was common.