How do I ask for eggs, sunny-side-up, but with the top cooked more thoroughly?

There are two different techniques possible here – obviously, simply frying for a longer time will cook the yolk more thouroghly. You might order this as “sunny-side up, well done”.

However, what you achieve by cooking with a lid isn't really considered just frying, but Basting, so what you want to ask for is probably a basted egg.

You can ask for eggs over easy.

This is when the cook flips the egg over once it's ready and allows the wolk to cook a bit on the hot surface of the pan. The phrase is an American one, if I'm not mistaken, but is probably understood anywhere.

If they have a heat lamp, ask them to hold it beneath the lamp closer than it would be on the counter for half a minute. Use it like a toaster, because that's really only way you can get it cooked to that level of detail. You don't want to cover and baste it, nor can you flip them. You could use a torch but that may overcook them or burn the whites.

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.