Yes, it might have been better to just say Primary (maybe - dead is after all an adjective in front of star - a corpse is also a dead human). And yes, it counts for the DM-1 if it's dead.
The potential complication is that for the Primary star, if you go down the default table path, a roll of 2- gets you the Special column, which doesn't include dead stars as a Primary - this matches the maps in Charted Space (except for a few purposefully placed objects), but not so much 'Reality'* where single white dwarfs - or pairs of white dwarfs, or the same for brown dwarfs, neutron stars, plus black holes, and mixes of the list - are often the only 'stars' (objects-formerly-known-as-stars?) in the system. For "Reality" as the text suggests, it might be more fun to use the Unusual column instead. And the Optional Variant on page 16 that will make many more red dwarfs.
*Or that version of "Reality" in which the universe is arranged in a single layer of hex-shaped cells.