"Over" = cooked on both sides, such that the whites are completely solid.
"Over easy" = cooked on both sides, but lightly enough that the yokes are still runny.
"Over hard" = cooked on both sides, long / hot enough that the yoke is completely cooked, pale yellow and dry like a hard-boiled egg.
"Over medium" is, as you might expect, somewhere in between. Cooked on both sides. Yolk is NOT runny; it is completely congealed. But it's not cooked long / hot enough to get the yolk completely yellow. It should be congealed but still very orange in color.
The idea is that you need to cook it enough, even the yolk, that you kill any pathogens; that usually means internal temp > 140 degrees Fahrenheit. "Over easy" and "basted" may not accomplish this; the centers of the yolks aren't there. The more you cook it, the greater the fraction of the protein that is denatured, reducing the nutritional value. "Over medium" is expected to be "just right," safe to eat but minimally denatured and still retaining maximum nutritional value.
There are an increasing number of people (or, at least, so it seems to me) who are getting really picky about their food. Many have weakened immune systems and can't handle "risky" foods. Some just want maximum nutritional value from what they eat and they're not satisfied with the old norms. It has taken some practice for me to get reasonably good at "over medium" because I live with one of those people.