Remaster: an ambitious port that updates aspects of the original game and cleans it up. Can have new music, assets, QoL features added to it, etc. but at it's core, it is that original game. It's, on an financial level, a smarter thing to go at when bringing a game to a newer audience but not wanting to just port it.
Example: Xenoblade Definitive Edition, Alan Wake Remastered and GTA 3 Trilogy which just released (in a poor state)
Remake: a game that is much more ambitious than a remaster, in that it can be as faithful to the original or as ambitious as possible. It is a completely different game from the original and more of a copycat of the original to varying degrees. The overall concepts, philosophy, plot, etc is recreated in this other game. The scope of that recreation is up to the developer. A game can attempt to be like the original or take liberties and add extra or take away something from the original source, but still follow the original plot in the overarching way.
Example: RE2 Remake, Shadow of the Colossus, Pokémon Brilliant Diamond.
Retellings fall under or are extension of Remake, except they just have the same characters and settings but the plot is not really like the original, this is what Final Fantasy 7 RE is.
That's *my* definition of course. Neither are bad.
Developers go about it their own way of how they want to do remakes and remasters. Nintendo is more faithful with their remakes, as to not alter a title away from the source. Square Enix was ambitious with its remake of 7, not really sticking to the original.
granted the original couldn't be stuck to really considering how ancient it was. :p