In A New Hope, the Death Star is only referred to as such (in dialogue) on four separate occasions; three of which are the intercom at the rebel base on Yavin 4, so I'm not sure they count as truly separate. It does appear in writing in the opening crawl and, likewise, in The Empire Strikes Back (where it is obviously not verbally mentioned at all). In Return of the Jedi, it appears again in the crawl and in dialogue -- to my count -- five times (more, but not by much given the plot). I haven't had a look through the scripts of the prequels; however, I would assume it's hardly mentioned, as it's still in the planning phase.
Anyway, it seems the only time it is referred to as the "Death Star" by an agent of the Empire is that one non-intercom-time in A New Hope and again by Darth Vader in Return of the Jedi. Was the "Death Star" therefore an informal name used by the Rebellion -- albeit sparingly -- that caught on with the Empire as time went on and, if so, did it have an official designation?