I call that “properly scrambled”. There are at least 1001 different ways to ruin scrambled eggs. In my experience, most people have no clue, and I include myself in that category many years ago. I make my scrambled eggs with cream cheese, which is og’s gift to eggs. Like you, once they begin to set, I add in the cream cheese, then I start folding and lightly chopping with the edge of the spatula. They end up more. . .piled. . .than anything else, and beautifully cooked. Start to finish, it’s probably no more than about two minutes, if that.
You scramble the egg in a bowl, usually with a little milk. If you want cheese, it can go in then. Then you pour it into the pan (medium heat). At this point, you gently move the egg around as it solidifies so that the liquid parts flow into the places where you slid aside the more solid portions. When the sheen starts to vanish, put it onto a plate (it will continue to cook).
If you like your scrambled eggs with mix-ins (what people incorrectly call an omelet), add them just after you add the egg to the pan. If they need to be cooked, do that first.
Back to the OP, a fried egg is an egg fried in butter. As people have noted, there are various ways to fry one.
I prefer over easy, but I’d eat the eggs however they were fried. I would have specified ‘over easy’ when I ordered. But unless the eggs just tasted awful, getting them in some other way wouldn’t be that big of a deal. The second time I went to the place, I’d be very specific.
An omelet isn’t scrambled eggs with mix ins. And butter is good, but so is lard, and other fats and oils as well.
What he said. I like my yolks cooked hard - so I ask for my eggs over hard. I usually get exactly what I’m looking for that way.
While I certainly wouldn’t object to butter, I would have thought that the default frying medium for eggs was bacon grease.
A properly scrambled egg is thoroughly mixed before cooking (white and yolk well blended together, often with a bit of liquid, usually water), then stirred constantly in the pan during cooking. The result has a uniform pale yellow color and a pebbly appearance. Most restaurants don’t actually scramble them, they mix them partially before cooking and stir them a time or two in the pan, making what I call a chopped egg.
I think traditionally a fried egg is cooked on both sides with the yolk a little runny–at least for a traditional fried egg sandwich
Concur.
A fried egg sandwhich is an egg cooked in a few tablespoons of fat (usually bacon fat). Yolk should have a hint of moisture and not dried out. Two slices of bread and mustard and you’re good.
Fried egg for breakfast? Means nothing. You got to specify how you want it. Sunny side up, over easy, well done.
Survey says: I like them over hard. No runny. Blech.
Well the cook certainly for not knowing that “over-easy” means you flip the egg. It’s in the name: “over”.
Technically, in culinary terms, the cook is nowhere near a chef unless he’s been to formal cooking school, so there’s that if one wants to nitpick.
You were closer to right, though a bit confused on terminology as well.
Over all, I’d say it’s losers all around. 
Question, how was the English of the waitress and especially the cook? For that matter, how is your English?
But folks admonishing him for not asking specifically what he wanted, read the story again. He specifically states that today, because of past experience, he specifically explained what he wanted to the waitress when he ordered. Then when it came out wrong, he tried again with the cook. So he can’t be faulted for assuming they’d come the way he wanted, he tried to be clear. Service in that place wasn’t very good.
I’d tally it as “find another convenient place that at least tries”.
whoa whoa… I think this kind of speaks to the issue of the thread. Mustard?
Never have I been served a fried egg sandwich with mustard. Looked at some recipes and it is a thing but I’ve never seen it out in the wild.
At most it’s mayo and maaaaybe cheese and/or tomato.
How can they get the terminology wrong?
Sunny side up means you never splash grease on the yolk. It’s bright yellow when served.
Over easy means flip it over for about 10 seconds to set the yolk. Or you can use a spoon to splash hot grease on the yolk. You get a white film over the yolk.
Well done means you flip it for about 20 seconds and use it for a hockey puck. 
Sorry, I really dislike over cooked eggs.
True dat. Scrambled eggs are mixed in a bowl (or in a pan, if you’re quick), but not beaten into a froth like eggs for an omelet. Omelet eggs are often mixed with a bit of water, scrambled usually with a bit of milk. A common mistake is to add too much milk; more than a tablespoon for three eggs is too much.
Me three, if somebody asked me, generically and casually, for a “fried egg”, I would just prepare them per the house custom and give them an over easy fried egg with a runny yolk. But it should be incumbent upon the waitress or counterperson to ask how you want your “fried egg”…over easy, sunnyside up, over hard, etc. And it sounds like he gave them to you default “over hard”, what you describe somewhat confusingly as “scrambled yolk”. As a fry cook it would never occur to me to give someone a fried egg over hard by default.
I will say, It is really bad etiquette and extremely bad business practice for the chef and countergirl not to have worked it out and given you what you wanted, however. Was there a language barrier problem or something… what is left unsaid here intriques me morethan the question. Why wouldn’t they fix you new eggs>?
My father used to take his eggs basted. Maybe one in ten cooks knew what that meant.
Usually, when I make a “fried egg sandwich”, I cook them “over hard”, so maybe the “fried” egg is a truncation of the fried egg sandwich, which I have seen as a mainstay on several diner or lunch cart menus dating back to the early 1900’s.
Thanks to the OP for starting a great thread that really added to my knowledge. I had no idea about the true definitions for fried eggs. It seems this can differ quite a bit due to local culture and custom.
Around here, when you order a fried egg, it’s usually assumed you want it “sunny side up”. Most of the larger chain restaurants have disclaimers on their menus saying they’ll cook the yolks well-done unless you ask otherwise and you do so at your own risk.
Now I know to be careful when I’m having breakfast away from home. Probably my best bet is to get the scrambled eggs.
Bri2k
I should also point out an important denouement, I purposely break or puncture the yolk, then fry “over hard” when I make a traditional fried egg sandwich. Or at least that’s the way I was taught.
Technically, that is not true. There are a lot of chefs who have never darkened the door of a cooking school.