Does this still count as "scrambled eggs"?
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...