The M1903A1 was an M1903 with a pistol grip stock, no matter who made it. The contract with Remington was for the M1903A1 (technically the current service rifle, although almost none were actually in service), but most were made with the "S" stock, without a pistol grip, or with the "scant" stock with only a slight pistol grip.
Remington's M1903 rifles were marked "1903"; the "A1" designation was never stamped on any rifle. Remington's first rifles were pure M1903. As time went on, Remington proposed changes for ease of manufacture, and eventually was producing what became known as the M1903 Modified. That was never an official designation, since all parts were still interchangeable with the original M1903. Finally, changes were proposed and approved for parts that were not interchangeable with those of the M1903 and the rifle was given the "A3" designation (and so marked). The sniper version, the M1903A4 differed in not having sights, having a bolt cut for a scope, and of course the scope and mount; the marking was still "A3." The "A4" designation was not marked, but the receiver markings were divided and moved to the sides so the serial number could be read with the scope mount in place.
On the Pedersen device, a note that while the 7.65 French Long cartridge uses a case of the same dimensions as the "U.S. Pistol Cartridge, Caliber .30", the French round can't be used in the PD because the bullet is longer and it won't fit in the magazine.
Jim