A machine gun is a crew served weapon firing rifle rounds. A sub-machine gun fires a pistol cartridge in a personal weapon. An assault weapon fires selectively for a less than full size round.
In between the two you had things like the M1918 BAR that was used as a squad automatic weapon. It wasn’t crew served and was fed from a 20 round box magazine, but fired the then standard .30-06 round like the 1903 and M1 Garand.
It was superseded for the most part by the M14, only some of which were issued with the fire selector installed, and in very limited use the M14A1 which had a pistol grip, a modified in line butt stock and muzzle device to better control recoil in conjunction with a bipod and a fore grip integrated with the sling to perform the role of a light machine gun.
However, the M14, given its full sized cartridge and select fire capability, would still be considered a select fire “battle rifle” like the German G1/FN FAL/L1A1 and the German G3 and various CETME derivatives.
The FAL as originally envisioned would have used the .280 British cartridge (.280” land diameter, .284” or 7mm groove diameter) which was an intermediate cartridge with a 140 gr bullet at 2550 fps.
We went with the 7.62x51 instead (147-150 grains at 2800 fps) and forced it on NATO, then almost immediately started developing the 5.56x45mm round, which was arguably a bit too far on the weak side (55 grains at 3250 fps).
Now that we are finally going with a 6.8mm (.277”) round we’ve swung all the way back in the battle rifle direction with the hybrid cased 80,000 psi 140 gr bullet at 2950 fps 6.8x51 round from Sig.
We’re slow learners who like to repeat our mistakes.