I have a 1903 A3 Myself, its a Superb Rifle, The recievers were stronger because they were made out of Nickle Steel, Some parts of the Rifle were made cheaper, like the stamped floor plate, trigger guard, barrel bands, and other small parts. Given a choice between the 1903 A3, and the 1903, I would prefer the 03a3.
During early World War two, there were not eneough of the M1 Garand Rifles to go around, so our government started production of the 1903 Springfield back up,but took some shortcuts, in their manufacture, and fitted them with appeture sights, which was what the troops that had trained on the M1 Garand were accustomed too.
You mentioned on the Art of the Rifle, that a friend gave you a replacement stock for your rifle but it was not the same, from your discripton it was a stock for a 1903, It will work, but your floor plate will probably not match up, Anyhow, You have a good friend there to bring you that, the other fellows on this forum, are alot more knowledgeable than me, and will tell you everything you ever wanted to know about the firearm you have in question. Good Luck.
Here is a not so good picture of my 1903 A3