1) Not all spirits are undead.
2) Ghosts are often indistinguishable from spirits.
Hence: either we implicate ghost to spirit and rework ghost to only refer to instances where someone clearly died, or we alias ghost to spirit and remove the implication from ghost to spirit.
Not to mention that Undead tends to imply "brought back from the dead", which is questionable even when the person is dead (Do people who ascended and became angels/summons/etc count as undead? If not, why should ghosts?)
Updated by Riversyde