your information is mostly correct but with a few addend oms,
the 1903 was pretty much retired after WWII, as was the newer, updated 1903A3 design. however the 1903A4, which was only made by Remington continued on until early on in Vietnam as snipers. the 1903A3 and A4 actually have little in common with the 1903 and 1903A1.
main differences are:
the front sight base are different
the rear sights on the 1903/A1 are leaf style mounted on the barrel while the A3 has a peep sight mounted to the rear of the receiver, the 1903A4 has the dove tail for a rear sight but is drilled and tapped for a scope mount.
the floor plate and trigger guard on 1903/A1 is a two piece of milled steel while A3s and A4s have a single piece of stamped metal.
since the A4s were designed to have scopes mounted to them, they had bent down bolts that wouldn't interfere with the scope, standard bolts couldn't rotate up enough to cycle with scopes in place.
the stocks change styles a lot but some themes seem to stay the same.
a Straight style with grasping grooves were found on 1903s while the 1903A1 had a "C" style or full pistol grip stock. the 1903A3 had a straight style without grasping grooves and the A4s had the C stock. a intermediate "scant" stock was easier to make than a C style so during war time the scant style was used to replace broken C stocks so it is not uncommon to find A4 snipers and A1 rifles with scant style stocks.
a 1903 mark 1 was made with a small oval carved in the left side of the receiver to accommodate something called a pedersen device that converted the 1903 into a semi auto shooting small 30 cal pistol cartridges but the device was scrapped before any were ever issued. no real difference between the mark 1 and 1903 other than the cut and mark 1 stamped on top.
that's about all the variant info I've got.