Does this still count as "scrambled eggs"? Being Chinese-American, I grew up on this stuff. The English name "Chinese steamed eggs" comes from the fact that it comes from Chinese cuisine, and that "steamed" comes from the word 蒸 in 蒸鸡蛋. The word 鸡蛋 refers to "chicken eggs", but the word 鸡 may refer to chicken or turkey. The word 蛋 refers to "eggs". The Chinese language has no singular/plural distinction, but contextually, we know it's eggs, not an egg, because we often use two or more eggs. 

During the food prep, the eggs must be beaten. Afterwards, some water is added to increase volume. Salt may be added for flavor. Then, the eggs are set in the steamer to be steamed. 

In Chinese cuisine, there is also a separate dish called 番茄炒鸡蛋, which involves tomatoes and scrambled eggs. 炒 refers to a cooking process. The Chinese definition of this cooking process is: "把东西放在锅里搅拌着弄熟". Basically, you put something in the wok and mix until ready or fully cooked. Often in Chinese-American restaurants, 炒 may refer to the process of stir-frying. In this dish, 番茄炒鸡蛋, the eggs are stir-fried, but I think the process and the finished result look indistinguishable from scrambled eggs. I mean, you really do scramble the eggs, dump the eggs in the wok, and then stir-fry the eggs until fully cooked.

Here's the tricky thing. The English word "scramble" seems to refer to the process of beating the eggs. If that is the case, then wouldn't Chinese steamed eggs be considered and classified as a form of "scrambled eggs"? Or should Chinese steamed eggs be considered/classified as "steamed eggs"? Or should this recipe fall under both categories - "steamed" and "scrambled"?
I would think the definition of 'scrambled' comes from being constantly stirred as they set, which gives the final texture; not that they start as beaten eggs.

I've heard of an egg simply dropped into shallow oil [just as you would to make a regular fried egg] then stirred in the pan to produce a similar if less homogenous result, also called 'scrambled'.

I think the letting out of the egg mixture with water, milk or cream is just to produce a fluffier result, sofer texture. I don't think they change the definition of 'scrambled'.

From your two examples, I would call the first - beaten then left to steam, closer to an omelette rather than scrambled. 'Steamed' covers the cooking method, but it doesn't tell anyone what the end result will be like. An egg poacher will steam an egg, but the result is a whole cooked egg, not a beaten one.
The second, stirred as it's cooking, you could consider a type of scrambled.
I don't think the definitions are firm enough to really quantify much further. Call them a Chinese omelette &amp; Chinese scrambled eggs, then a 'western' audience would be expecting something similar but not identical to what they would normally have for breakfast.

The definition of the word scramble, if you scroll down to transitive verb, 2a &amp; b, is 


  to toss or mix together in confusion : to throw into disorder


and more clearly  


  to prepare (eggs) by stirring during frying


I consider that to be the defining distinction for scrambled eggs, separate from whether it is initially beaten or whole - the scrambling is what takes place at the time of cooking.

Having read the Wikipedia page, I'd be concerned that a British audience would be very confused if that was referred to as a 'custard'. Even qualifying it as an 'egg custard' a UK audience would still expect it to be sweet. Brits, almost without fail, think Bird's custard is custard.

Late thought: If it's already known in the US as "Chinese steamed eggs", then that's what people will have come to expect.
A rose by any other name...