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First of all, there really isn’t all that much story to the original Doom. There are a couple of paragraphs in a readme.txt file tucked away in the installation folder, and an ending screen after each episode… but that’s basically it.
As I understood it, Doom 2016 is a re-imagining of Doom, so the universes are not canonically linked. Kinda like how The Thing From Another World (1956), which takes place in Alaska, isn’t canonically linked with The Thing (1982) which takes place on Antarctica.
Sorry but you’re not the arbiter of worldwide speech, so you don’t get to decide that. The name of the 2016 game is just Doom. If there’s a chance of confusion you can add (2016) for clarity, as I did, but it really has the same title as the original Doom game and there’s nothing wrong with just calling it Doom. Which it is.
See:
It’s like Scarface and Scarface, or Cape Fear and Cape Fear, or Dune and Dune, or … you get the point.