A fantastic gun. In my definitely unexpert opinion it looks correct. If it is an original 5" 1950 44 Target it is worth far more than what you paid. But you will have to letter it to be sure.
Here are my two 5" 1950 44 Targets. Neither is original. Both shipped as 6.5" guns. Letters say one shipped to New Orleans and the other St. Louis in 1955. Story I got was that in 1970 they were together with one owner who sent them back to S&W for modifications. One had the barrel cut to 5" and the other had a replacement 5" barrel installed. Maybe the last one in inventory since they cut the other? The front siights look slightly different with the cut barrel having a sharper slope to the siight. Smooth triggers also installed. The cut barrel has the correct serial (how about the subject of this thread, number there?) indicating it is a shortened original tube. The other has no number being taken from the parts bin and not coming on a gun.
Note the lettering on both is fairly centered and nearly identical to one another. It appears the stamping on a 6.5" barrel was in the same location as a 5" barrel. Both of mine have the S&W logo on the side plate. As someone noted the left side location may indicate a correct short barrel version.
I really hope the OP's revolver is the real deal. It looks great and even if not original still worth what he paid and a great shooter.
Cut barrel:
Factory original 5" replacement barrel: