*golems (if it can work for a modern day culture's lore, it can work for you)
*constructs (the base creature type used already in the monster manual, just calling them by the generic term rather than giving them a specific creature name)
* constructicons (though this one may get giggles from people who know Transformers well)
* or don't give them a name as a whole, but rather they are simply considered Automatons that have souls and therefore individual names. but in the grand scheme, they are still just refered to as Automatons like the soul-less ones.
* Stonemen / Metalmen
* (Made up name) that is the same name as a historical figure (kind of like calling them all 'goliaths' to represent them as these massive men ... but whatever would be a campaign-world equivalent) -or- the made up name is the name or reference to the one who originally created the first warforged
* "Smarter Automatons"
* Automatons v2
* "The Crafted"
while those ideas may/may not be great, it might spark a better idea from someone...