My memory of Rebirth of Mothra is fuzzy to say the least (I should probably rewatch it sometime), but Death Ghidorah is obviously some sort of relative to King Ghidorah - but something slightly distant, like a cousin or something. It's possible that Death Ghidorah's looks stem from a variation in the Ghidorah species, kinda like how Irys was related to Gyaos (though this Ghidorahs and Irys/Gyaos analogy might be weak).
It is entirely feasible that Death Ghidorah came from the piece of Cretaceous King Ghidorah's tail, though I'm guessing the idea was that the King Ghidorah that Armoured Mothra fought at the end of Rebirth 3 was a sort-of clone of the original Grand King Ghidorah.
I have a theory that Death Ghidorah arrived on Earth first and tried to do what ultimately King Ghidorah succeeded in doing, only the Mothras (Mothrae?) of that time managed to stop and seal Death Ghidorah away. As I recall, part of Moll's dialogue in Rebirth 1 mentioned that only Mothra could stop Death Ghidorah, like she had done millions of years ago, to which Lora replies that there were many Mothras back then.
This is all wildly speculative from here. So, the Mothras of the Cretaceous (or whenever - I don't recall there being an exact number to the first ever Mothra/Death Ghidorah fight) managed to beat Death Ghidorah - possibly as Imago or perhaps even in Larva form. If they were all Imago, it's possible that after Death Ghidorah, they laid eggs which gave birth to new Larvae (which could have been the ones in Rebirth 3 who coccooned Leo). However, King Ghidorah arrived and the Larvae were no match for him, and thus KG decimated the dinosaurs. This all changes, of course, when Leo travels back in time to the Cretaceous period and pretty much kicks the tails of the more junior King Ghidorah, having fought the adult in the present.
I guess my whole theory in a nutshell is that Death Ghidorah was first but got beaten, and then King Ghidorah showed up and did what his predecessor failed to do. And speculating wildly is kinda fun, I must admit.
But feel free to correct me at any point.
EDIT: Looks like Musuko beat me to it. Ah well, my wild theorising got my brain warmed up, if nothing else. 
"Crossing into established events is strictly forbidden. Except for cheap tricks."
- The Tenth Doctor, Smith and Jones