Unless there's an obscure cartridge that I can't remember right now, I think you're referencing the same thing. A No 4 Mk1 is chambered in 303 British. There may be some headstamped Enfield though I've not seen any. What I've got here is just 303 and then date stamped. It was used in the Enfield series of rifles and I think in one of the belted machine guns --- again, poor memory.
I had a No 5 for several years and always bought 303 British ammo for it. Never a problem.
Hopefully, someone else will jump in here and either confirm or say "no, dummy....." with the right info.
303 Enfield first refered to the converted Martini's and then to the very first Enfield Bolt guns, as they where made at the Enfield armory,, then during WW-1 with the advent of the 1917 Enfield it confused the issue a little ,, as the Americans used it in 30-06 and the Britt's in 303 British,, so most Americans just short'en it to 303 Britt, and most ammo compianes in later years followed
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