Ovinomancer
No flips for you!
Bob. Or maybe Robert if you're being formal. Unless of course their name is really Margaret and for some reason people call them Peggy, which is something I've never understood. I mean, seriously? It's like calling Richard "Dick". The only commonality is a random vowel.
Anyway, I digress. Warforged works because it's simple and evocative, unique without being totally made up. They were forged for war, it's right there in the name. So maybe you need to start with their origin story. If they were not created to be warriors, why or how were they created? Are they still being built, and if so why?
Barring some bolt from the blue, you could just use a standard term like androids, automatons, mech or from sci-fi synthetic humans aka synths. Robot for example comes from Slavic robota for "forced labor".
So in my world I might use google to look up "slave labor" in latin to give me "servi laborant" which could become Serborants or Serlab? I don't know, neither really rolls off the tongue but you get the idea. An option is to just make up a new word that you think sounds cool, it doesn't need to mean anything. Maybe they're just named after their inventor? Kind of like how we call Frankenstein's monster Frankenstein?
Other options would be more how they were made. Soulforged, Constructed, Metal Men, or just Forged Ones.
But seriously? Jack for John? Really?
You probably weren't looking for it, but most of these are due to the English and their rhyming games. Those not due to English are the fault of the French invading the English.
Richard is obviously straightforward, Dick working two angles of fun there. Peg as short for Margaret is Margaret > Maggie > Meg > Peg. The middle bits there are questionable as to which came first -- it's possible Meg showed up and became Meggie/Maggie. Regardless, at some point there were enough people swapping the M for a P for it to stick. If you really want to have a fun one, "Daisy" was also a very common nickname for Margaret. That's because the French word for daisy is marguerite, which sounds similar.
Jack is more fun, and has to do with the French invasion of England. Johannes was the root name, which was adulterated to Jehan and Jan. The Middle English loved the suffix -kin, so Jehan and Jan became Jankin. The French nasal pronunciation, picked up by the English at the time, smeared this into Jahnkin when shortened to Jackin and then Jack becomes apparent. Reportedly, this has nothing at all to do with the Francophile Jacques with is James in English.
Ned and Nan as short for Edward and Anne are also from Middle English, which used "min" instead of my, so when speaking of your little Ed and Anne, you said 'min Eddie' and 'min Anne'. A few short centuries later and the N had left my and moved to the names.
Ah, the things you pick up married to an English major with a love of the history of the language. I knew the ones about Dick, Bob, Jack (suprisingly) and the Ned/Nan added N's, but that also meant I knew were to go about Peg. The Daisy bit was completely unknown to me before I looked into it.