"Trusting your eye" is really as many methods as there are people using it.

the first two methods sound as idiosyncratic and arbitrary as the last. (... I say without knowing anything about them)

Your students are obviously referring to the optical middle. :o)

my method isn't really very helpful: "It doesn't have to be right. I has to look right."

I haven't heard of an "optical center" other than on the horizontal axis, for spacing purposes. That was championed by David Kindersley. I think there is something to his idea, but he never really nailed it down to a formula that worked consistently.

Vertically, when you divide a space vertically such as e or E, the visually equal division is somewhat above the geometrical center. But I suspect that that apparently equal division will vary with the design of the rest of the figure. So, eg the crossbar of the H and A will be in different places. So there isn't one single number for a "vertical center."

I find the geometric center and then trust my eye to work out where the optical center is. It gets a lot easier after the first five hundred times you do it.

It really depends on the distribution of weight in the character, in my opinion. Optical center of the /e/ would be different from the optical center of the z with a stroke through it.

Let's scope on the simplest example, because we all are aware of slight differences caused by interaction of more complicated shapes, like letters.
I am not interested in optical centering itself - it's not my life's goal - but lastly I am really curious of what causes we need to place something that "has to be in the middle" at least a tad ABOVE. Is it related to - I don't know - neurophysiology (like anticipating an event, eg falling object)?
The Evans' rule seems to be linked with the golden ratio; it creates strong diagonals feeling, while Sutton is probably more an "optical tricker".
Maybe the relation among the four examples is rather culture- or design-driven thing?

Maybe I'm not young, healthy, and surely not nice enough, but the last example feels unbalanced to me. Obviously it depends on context and application, but I probably would go with something inbetween yours and the last one.

So you are the second youngest, nicest and healthiest.

PS: Indeed, I should put the square two pixels lower.

I agree with Riccardo, but the important thing in designing a typeface is to make the divisions work *in conjunction with other letters*. All of these could work, depending on the design. I'm sure you can find the cross bar on the H at all of these levels in good fonts. That's why the vertical alignment always is a matter of the eye, looking at, and deciding on, the whole design.