In the radio drama, Death Star is merely a project code name, not necessarily the official name. I kind of like thinking of it that way: it's a code name that the Alliance adopted for propaganda purposes. It seems to fit your interpretation perfectly. But for the record, I don't think the Empire cares much at that point if they are offending their population with scary names. They intend to use "fear [to] keep the local systems in line," not subtlety, as Palpatine did* to first gain control.
*cough, yeah, Palps was reeeeeal subtle as he assumed power...I can totally see why no one suspected him.