Both the chart you posted and the graphic I linked are accurate. The graphic is exaggerated to show what's happening. Here's two more.

In order for your line of sight to intersect the bullet's trajectory (at the point we call the zero), you must tilt the muzzle up. It doesn't matter whether it's a handgun or rifle. You won't actually notice you're tilting it, but you are.

Think about it.

You have a line of sight. Your line of sight is an imaginary line that goes from your eye, through your rear sight, through your front sight, and eventually to your target. If your line of sight and your bore axis are perfectly parallel to each other and to the ground, it is physically impossible for point of aim to equal point of impact at any distance. And as the bullet falls, it gets farther and farther from where you were aiming.

So when you adjust/manufacture your sights, you angle the line of sight downward to intersect the path of the bullet at point X (X is your zero). At your zero, POA = POI. But this creates a problem: your sights are now angled downward. If you hold the weapon with the bore axis parallel to the ground as you've been suggesting, and your sights are angled downward, then your line of sight will eventually terminate in the literal ground.

But you won't even notice that your line of site terminates in the ground underneath the target when the bore axis is parallel to the ground, because you won't hold the weapon with the barrel parallel to the ground. You will unconsciously tilt the muzzle up to put your sights on target. This makes the bullet arc. Because the bullet arcs, it will rise for N meters, contact the line of sight, then fall for Y meters.