To me, this is a remake:
Movie A: characters, story, plot, direction, setting, theme.
Movie B: comes out after movie A, has the same characters, the same basic story, is not a sequel, does not acknowledge movie A at all, same basic theme with the same characters.
Nightmare on Elm Street for example. New one had the same basic story ( for Freddy) and same character, same basic plot, but new setting, new characters (different kids), new ending. It was a 're-boot' but also a re-make as the original character and story seed was still there.