there appears to be a sleeve with two grooves near the muzzle of the barrel on the '03 in question.
That is just the way the barrel is turned and grooved for the front sight base. The A4 used A3 barrels with all the cuts, just that the iron sights were not installed.
There are THREE main 1903 stocks - not counting the different handguards for 03 and A3. The S stock with straight grip was the original issue stock from about 1905 on. The C stock with pistol grip was adopted in the 1920s. A 1903 with a C stock is a 1903A1 but only the Remington A1s are so marked. The stock "cut so you could get more stocks per blank"
is known as the "Scant Stock" with a faint pistol grip. More to hold on to than an S stock, less wood than a C stock. There are a lot of minor variations, Boyd says they have seen eight different variants.
CMP has C stocks
Also Boyds and Sarco.