If I ask for eggs, sunny-side-up, they will have yellow barely cooked yolks.  At home I avoid this by placing a lid and creating hot air or steam that cooks the top more thoroughly.  How do I ask for this style at a restaurant?  What term do I use? 
As you and others have observed, there is no unambiguous yet precise set of egg vocabulary for you to use here that will be understood generally, especially in countries where it is not common for people to have very specific egg requests in restaurants/cafés/diners/etc. Instead, I would advise you to politely explain your preference when you order, in much the same way as the title and body of your question: 'I'd like my eggs fried but with the top cooked more thoroughly rather than runny, if that's possible'. The cooks are the food preparation experts here and they know what equipment is available to them, and this avoids linguistic issues or confusing them by demanding a specific method of achieving your goal. 