See that little pipsqueek on the far right? That's what comes out of the "assault rifle" the killer from this story used. It is literally the size of 3 BBs glued together, with a few grams of powder behind it. Cartridge and all, it weighs less then 20 grams.
The round used by the primary firearm in this incident was, according to all the reports I've seen, the .223, aka the 7th round in your chart labeled "5.56 x 45 Nato." There's an important digit of distinction between a .22 and a .223. The projectile is only slightly wider, sure, but it has a lot more mass (approx double) and is moving at a much higher velocity (around 900-940 m/s vs 300-500 m/s).
For pedantics, .223 and 5.56 are not the exact same thing, but they're externally the same, i.e. the same size cartridge (casing and bullet). The interior volume of the 5.56 round is smaller, and can create slightly larger chamber pressures when fired in a gun designed solely for .223.
Not that it matters, but I'm a certified firearms instructor (certified to teach pistol, rifle, home firearm safety and personal protection) and strong advocate that the concepts being banded about for "gun control" are absolutely ludicrous at best, but that's not relevant to this specific bit of misinformation.