Let me simplify this:
Remaster - To edit again an already existing work.
Remake - To recreate from scratch an already existing work.
Xenoblade Chronicles Definition Edition is an edition of an existing work, fitting your top definition.
Final Fantasy 7 Remake is a recreation from an already existing work from scratch, fitting your bottom definition.
But to be more precise, a remaster isn't just an edit: patches, mods, and ports are also edits. A Remaster is an edition fit for newer generations of hardware. They generally replace outdated engines and graphics with ones appropriate for the generation.
Remakes are more precisely defined as new adaptations of an existing game in the same format (Film remakes are adaptations of earlier films, video game remakes are adaptations of earlier video games).
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