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:


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.
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.