I don't think that's right, though I do agree there are different levels for a remaster, at the end of the day they are a remaster. Many of the things OP listed are not a remaster. This is how I like to see it:
1. Port/Emulation. The lowest effort. It's essentially the same game, just being made to run on newer platforms, perhaps with some minimal improvements, like running on a higher res or better framerate. Maybe some new controller schemes. I think of the many ports the Wii got, or even the Switch, such as Bayonetta 2.
2. Remaster. Medium effort. Build upon the previous level, where the dev spends significant resources creating new textures, assets, improvements to engine, perhaps fixing egregious bugs and adding quality of life improvements. Adding new features and modes are on the table. Metroid Prime Remaster is here.
3. Remake. Highest effort. Some of the original code might still be present, but the game is essentially being built from scratch. It's essentially a new game, but it respects and pays homage to the source material. Generally speaking, the story and/or "feel" of the game are preserved. REmake and RE2 Remake come to mind.
4. Reimagining: This goes beyond a remake. It's a completely different game, doesn't attempt to preserve the original experience, but rather provide something new in a different direction. Story, characters and settings might greatly differ. A completely different game by all means. Best example I can think of is Silent Hill Shattered Memories.