When Death wins the 200 men of the ship's crew, they all die a conventional death. The mariner describes this death as follows:

With heavy thump, a lifeless lump, They dropped down one by one. The souls did from their bodies fly,— They fled to bliss or woe!

It is clear that these two spectres that appear on the ship and play a chilling game of dice for the life of the Mariner represent what their names say they are. Death represents complete death, and Life-in-Death represents a state of death that exists in life, that the Mariner has to suffer because Life-in-Death wins the Mariner, whereas Death wins the life of the sailors, resulting in their deaths. Note how they are described:

Are those her ribs through which the Sun
Did peer, as through a grate?
And is that Woman all her crew?
Is that a DEATH? and are there two?
Is DEATH that woman's mate?
Her lips were red, her looks were free,
Her locks were yellow as gold:
Her skin was as white as leprosy,
The Nightmare LIFE-IN-DEATH was she,
Who thicks man's blood with cold.

Clearly focusing on their descriptions, and in particular the description of Life-in-Death, shows they are deeply suggestive of what these figures represent. The fact that Life-in-Death wins the Mariner, and the way that we are presented with how the Mariner lives--doomed to wander the world and share his story thanks to an uncontrollable compulsion--shows the life-in-death that he has to endure as his punishment.