Classification of Story Relationships

Sequels
Direct continuations of a previous story.
Prequels
Direct precursors of a previous story.
Adaptations
Conversion of a story to a different medium.
Remakes
Re-writes of an existing story with the implication of accuracy.
Reimaginings
Re-writes of an existing story which explicitly diverge.
Re-write/Retelling
Re-writes of an existing story without implicit accuracy. Though this can also be considered a meta-category for remakes and reimaginings.
Stand-Alone Remakes
A stand-alone or spin-off which takes inspiration from an existing story but which makes no promise or attempt to remain accurate. Accuracy is coincidental.