What determines or controls the manner in which a structure is made? Is it the race flag, the Construction field in "Units.dat -> Graphics tab -> Sprite Properties", or something else?

When I remember right are in Datedit all graphics, but I think you have to make it with firegraft
"Building Morph" "Wapr In" and/or "Construct building"

Warning: If you make a special building with terran construction it will crash the game when you don't iscript another graphic! [the "almost done" frame is missing at special buildings!]

Uh... I think it is on firegraft...

In worker's button sets, u can choose to make a building the zerg way, or the terran way, or toss way..

But u still need iscripting to make them work thou, if u wanna change it

Ah, okay, you guys are right. I see it. Thanks! I was fairly sure it wasn't the race flag or graphic, but couldn't remember what was missing.

To expand a bit more, it is determined by the build type action called.
You can make for example, a barrack be built by a scv and with "protoss build" order.
Additionally you'll have to give the scv the proper unit.dat requirement order to use such build type.

Also, the AI doesnt like you changing build methods. Terran AI will always use the Terran way, Toss Ai will always use toss way, Zerg AI will always use zerg way. Keep that in mind if you plan to have AIs playing the mod. If you dont have the correct iscript animations, it will crash when the AI tries to build.