The remaster MUST retain the original story line. Adding new chapters or changing parts can chase away the core fans. Updated graphics, music, and controls can be awesome, but consideration for the original story and how those graphics and controls can change the story needs to be looked at. If the original controls had every turn a full 90°, updated controls may make that unreasonable for a puzzle; changing that puzzle could destroy how the game feels so maybe the controls should enforce tight turns.
A re-imagined game is absolutely fine, the modern Tomb Raider games are great. A re-imagined game works because it doesn't stomp on the original, it steps aside and makes it's own path. A re-mastered game is the original and as such must keep the original footprint or it isn't paying homage and is instead crushing the trail made before it.
A variation of re-masters I love are the Mesen HD packs. The Metroid HD and The Legend of Zelda - Remastered packs are phenomenal and do nothing but place updated graphics and audio on top of the original game.