That's the idea. However I take issue with that approach, to claim that the only way to get a DA is to kill and demon and that it will mutate into weapon isn't entirely accurate, it's not the only way to create one. DMC1 had you collect these weapons from the stage and those weapons don't have anything that implicate that sort of origin.
Actually the backstory of the 2 Devil Arms in DMC1 (Ifrit and Alastor) hints to them originally being demons but are now weapons and even when Dante got the 2 weapons he didn't just pick them up quite easily when he picked it up or try to pick them up they came to life and tried to kill him (Alastor impaled him and Ifrit tried to rip him apart) and Dante had to overcome these weapons to actual start using them.
So one (Capcom) can easily alter these Devil Arms in DMC1 to fit into the same context as DMC3 and DMC4 such as by stating the DA in DMC1 were once demons (cause it said they were or at least gave them titles such as Alastor the Thunder Devil and Ifrit the Fire Devil to refer to them as more than just random demonic weapons but living demonic weapons) who were turned into DA but by someone else or some other unknown reason and the reason why they tried to kill Dante when he received them was because he wasn't their master (or they have no master) and this adds fuel to that one theory if someone else tries to use a DA that Dante obtained from a Devil it might reject them and try to kill them.
Plus DMC1 hints to these weapons once formerly being demons and being alive but adds zero details on how that is and why are these weapons alive or how demons can become weapons.
Also the DMC3 manga, does mentions the power in a demon comes from their name and Sparda is able to strip a demon o their name/power so that could hint to how the Devil Arm's work since the Devil Arm still retains the demon's name.
Then there is the question of the Sparda Trinity (Sparda/Force Edge, Yamato, and Rebellion). I can understand Sparda since Sparda just harnessed all of his own power in the form of a Devil Arm or Sparda's power came from his Devil Arm...never mind I don't know how the Sword of Sparda work and then there is Yamato and Rebellion. The games never explained how Sparda got Yamato and Rebellion or how they were made. I'm not sure about Yamato but I guess Devil Arms can be artificially made in Rebellion's case or something.
I guess the Sparda Trinity works differently from the other Devil Arms.