The Lee Enfield was initially an American design. It was issued to marines in I believe 6mm. Then the Britts got in and it was a Lee Metford. Then became the SMLE or Smelly. The .303 was an intermediate round. And the SMLE did have a good reputation as a high rate of fire. The Japanese based or basically copied and took off the rim for the 7.7 Jap.
The P-14 Enfield was originally designed for a new magnum length cartridge. I remember a .276 caliber or such. Due to the onset of WW1, it was redesigned for the .303. It was based on the 1903 Springfield, with SMLE and Mauser influences. Remington used the action as a sporter rifle and improved it over the years. The US used is as a service rifle in WW1 and WW2, the P-17. Actually, more were made than the 1903 and 03/A3 or 03/A4 Springfield.