Can I resolve to display only the: Death Notes GUI? Which, of course, everyone sees.

You can try setting the chat and console messages to false and see if that makes any difference .

 "Chat Format": "<color=#838383>[<color=#80D000>DeathNotes</color>] {message}</color>",
  "Chat Icon (SteamID)": "76561############",
  "Show Kills in Console": true,
  "Show Kills in Chat": true,
  "MessageRadius": -1,
  "Use Metric Distance": true​