How do I convey an order for over medium eggs without runny yolks?

Not to be dismissive, but this just sounds like a difficult guest to me. I think you'll find that this happens from time to time and it can be tough to know exactly what they're asking for.

There are two major possibilities here:

1. She was looking for the standard definition of "over medium", but has gotten under-done eggs before and thought she was clarifying; instead her additional information ("not runny") just confused things.
2. She's got some non-standard definition of "over medium", in which case she's going to find it difficult to locate any kitchen that can deliver. This is a bit like defining a "medium-rare" steak as one without any pink in it, then getting mad when you receive a well-done steak.

From a server's perspective, it sounds like you tried a couple ways to deliver what she asked for. Her not being satisfied with that could be the result of the kitchen not quite meeting your request, her not really knowing what she's asking for, or other factors entirely. Perhaps you addressed her in a way she found disrespectful or annoying, and used her eggs as a proxy to complain.

(A side note from my experience: somebody insistently telling you up-front that they'll send back their food is a huge red flag. It almost always means they're looking for an excuse to cause a fuss.)

Frankly, it's not unusual for someone having a bad day to take it out on service staff over something minor. It's an occupational hazard. When this happens, be polite and clear with your manager about what happened and explain the situation as best you can.

My understanding is:

• Over easy - yolk runny and some white runny
• Over medium - yolk still runny and white firm
• Over medium well - yolk not runny but still soft and white firm
• Over hard - yolk firm; break the yolk before flipping

Talk to the cook and ask them what to call a not runny yolk.

You could just have a cook that is not good with eggs.

Based on a comment there seems to be misunderstanding about talk to the cook. You don't ask the cook what it is called to argue with the customer. You ask the cook to know what to write on the ticket. If someone orders a not runny yolk what do I call it on the ticket. If someone orders a well done steak but with some pink you don't argue that is medium well. Just turn in a ticket for medium well and scratch out the medium before you give the ticket to the customer.

"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.

The guest should have ordered - fried eggs "basted medium". This is where eggs are fried for 1 minuted and then hot grease ladled or spooned over the yolks until the top of the yolk is cooked as fast as the skillet side. The eggs are never flipped and the yolks are never broken. Basted eggs are removed from the skillet while the center of the yolk is still liquid in the center but hot. To prepare fried eggs - basted requires the cook to stay focused on the order. Most cooks are tossing too many orders at one time to do basted eggs justice. If the guests seems a bit particular with their eggs, politely recommend the guest order scrambled eggs. It works every time.

I know this because..... I was married to that woman.... and she taught me how to fry her eggs.

Over medium should have a fairly firm yolk. Over hard is something entirely different (you break the yolk and mix it in). This is by definitions I've read at least. I have only seen one restaurant that does it correctly though and most seem to make their over medium identical to their over easy.

"Over medium" is that stage at which the yolk is still bright/deep yellow and not brittle, but not liquid/runny. Might say it's "thick".

Over-hard eggs have a pale yellow, solid, brittle crumbly yolk, like that of a hard boiled egg.