Optically is more center of mass, but ideal more 'center of visual weight'. Larger things are 'heavier', darker/more contrasty things are 'heavier'. Rounded things should extend over an alignment line a bit, flat shapes butt up against it (e.g. in the letters lO the O extends over the baseline slightly compared to the l, to visually connect with the baseline more).

In general the way it works is you align with whatever software says and you look at it and say does it actually look aligned or need minor adjustment? Because the point is to look aligned rather than to be mathematically correctly aligned. Like our letters above, they look more visually on the same baseline when they're not, then when all the rounded base letters sit most 'correctly' on the line.

The author is suggesting you use the mean X value of all the pixels in the shape, instead of the mode x value.

I'm sure fancier image editing software does this, but your eyes are pretty good at it too because it's just dividing the surface area in half, or thinking about it as the "center of mass" as you put it: where could you attach a string to this shape such that it doesn't tip over one way or another?