Difference and meaning between "remastered" and "remake"
Solution 1:
Although it seems like the terms are often used interchangeably within the industry and marketing, there IS a technical difference between the terms:
A remake typically shares very little of the original assets and code with the original game, distinguishing it from an "enhanced port", partial remake, or remastering.
source: https://en.wikipedia.org/wiki/Video_game_remake
In essence, the difference comes down to just how much of the original game code and assets are used, with a "Remastered" title only updating graphics, whilst a "Remake" could, potentially, include radical changes in gameplay.
An example of a game that has had its graphics redesigned is Halo: Combat Evolved Anniversary,[11] while the core character and level information is exactly the same as in Halo: Combat Evolved.[11][12][13]
source: https://en.wikipedia.org/wiki/Remaster#Video_games
To use the Bethesda example, Skyrim with updated graphics only would be considered a Remaster, whilst (and good god I personally want this to happen) a version of Morrowind with both updated graphics and a Skyrim-esque combat system would be considered a Remake.
Hope that helps.
UPDATE: Having asked a friend who used to work within the industry (on several remastered titles), he confirms this is the way it is supposed to be used. He also said, however, that even within development and testing the words were often used interchangeably both verbally and on documents.
Solution 2:
TL;DR: remaster=close to original, remake=close or far from original, depending.
These words are meant as analogies to the equivalent in music and films. In music, a remaster specifically means to put together a new "master copy", a recording from which all CDs, Tidal tracks etc will be made. When a song was produced, the vocals and instruments may all have been stored on different tapes, before being edited and processed into one single "master". remastering redo this job.
A "remake" on the other hand, is a new and independent recording of a film, based on an earlier film. When this happens to a brand, such as star-trek, we talk about a "reboot".
In games, a typical example of a remaster would be the new releases of Monkey Island. Gameplay-wise they are exact replicas of the old games, so much that you can switch between new and classic modes at a keypress. A "port" of a game to another platform would also be a remaster. An example of a remake would be Prince of Persia, where the basic concept of a jumping and slashing guy dressed as lawrence of arabia was kept, but the storyline and gameplay otherwise altered.
So.. Crystal clear? Unfortunately not. :-) These words are just analogies that people do with as they please. On wikipedia the article Video game remake use "remake" as I've used "remastered" here and use "reboot" for what I've called a remake. (Being part of a franchise is why you remake old stuff, hence the "reboot"="remake")
You can see the confusion in that Wikipedia has a List of video game remakes and these include several games with the word "remastered" in them, such as "Day of The Tentacle - Remastered" and "the last of us remastered".
Solution 3:
"Remaster" would mean they took the original game (code+assets) and improved it somehow, usually by improving the graphics and/or adding compatibility with new hardware/OS, but possibly also adding features.
"Remake" means just what it sounds like, they completely remade the game from scratch.
The same applies to things like movies and music also.