I think reboot and origin story used by both Capcom and Ninja Theory is what is making DmC have a lot of issues.
A reboot for me is completely rewriting the story and probably ignoring the old story because it was poorly written in the past, like having too many plot holes, stuff not making sense, was incoherent, etc. and just making the new story more understandable. It might involve an entire take on the story itself. Basically for me it's like the company saying "We screwed up. Let's try again." The DMC series for me is not on this path and DmC doesn't feel like a reboot. An example of a reboot would be the Transformers Aligned Universe began by Transformers: War for Cybertron.
An origin story is exactly what it is: an origin story. It's like Batman Begins of the Dark Knight trilogy (even though it was in another universe; Christopher Nolan's). Origin stories tell the origin of the protagonist, antagonist, conflicts, etc. But it is not necessarily a reboot. DmC is definitely not an origin story to DMC. It is, however, an origin story to Dante of DmC.
To me, DmC is more akin to the Noir and Ultimate (and the other) universes of Marvel. Noir Spider-Man had and used guns at times, while Noir Wolverine I think basically gave up and left his life of being an X-Men. Ultimate Spider-Man *SPOILERS* dies and is succeeded by Mile Morales as the new Spider-Man. Ultimate is closer to the Mainstream Universe, but Noir is very different to the Mainstream universe, which is what DmC feels to me: close but not close enough to the DMC.
Also, please note, I'm not that knowledgeable about reboots, origin stories, re-tellings, etc. I just hear them a lot and find the meaning from what happens; something that happens a lot and people aren't aware of.