The design of the bullet has nothing to do with it's classification. If you were correct, the P90 would be considered an assault rifle as opposed to a submachinegun/PDW. Hell if you made the FiveseveN fully automatic then by your definition it would be an 'assault rifle'. Three big features that make a firearm an 'assault' rifle are 1) Intermediate cartridge (i.e. smaller than 'full size cartridge') 2) Select fire and 3) Smaller/more compact than a 'battle rifle', though admittedly #3 is a rather archaic benchmark these days. Magazine capacity has no real classification purpose. Vietnam saw 20 round magazines for the M16's. They surely were assault rifles.
And he didn't ask about the M2 carbine, he asked about the M1 carbine. If the M1 Carbine was fully automatic, it would've truly been the first assault rifle.