A replaced stock is not at all unusual on a Smith gun that has been used, as this example obviously has. The engraving does appear to be any early vesion of the Trap grade pattern (all these patterns varied to some degree over the years of production; and I have observed at least three different variations of the Ideal pattern for what that may be worth). Here's my conclusions, although the photo is not great the "T" grade stamp does indeed appear to be the factory grade stamp for a Trap Grade gun and does not appear to be an over-stamp; but the fact that both barrels carry the Crown Steel logo is, for me anyway, the primary key to grade verification. Crown steel barrels were not used on grades lower than Trap; so I believe your gun to be an authentic Trap Grade. As to the records descrepancy, check with Cody one more time to see if a mistake has been made; however, in the final analysis the remaining records are what they are, correct or not correct; and the people at Cody have no control over what is recorded therein, or remains today other than actual preservation.
One final note, although once again your photo is not very clear; your post-1913 gun appears to have a two-piece top strap (if I am not mistaken I can see the screw slot). The question of when this feature was dropped from production was asked earlier this week; and although I was almost positive it was dropped sometime between 1913 and 1918, I did not respond as I could not recall a specific example. I believe Mr. Houchins posted that this feature was eliminated in 1909; but I believe it was the bushed firing pin feature that was eliminated around 1909, with the two-piece top strap being dropped at some point between 1913 and 1918. Thanks for sharing your gun. Tom