How Should Video Game Remakes Be Done?

I definitely agree with your definitions there, and I definitely like the latter; remakes that go above and beyond. Let's use the first 3 Generations of Pokémon ramkes as examples as to why I love remakes that go above and beyond.

So if we start with Fire Red and Leaf Green, they're the GBA equivalents to Red and Blue/Green. In these games you follow the story exactly, with a few side-quests thrown in and very small changes made here and there, but overall they're the same game. However, they did add a little bit of post-story in the form of the Sevii Isles, which does give the player a bit of new story to go through after the main story, but it's only a little. This makes me fell that FRLG falls under the first category; fresh coat of paint.

Next, we have Heart Gold and Soul Silver, the DS equivalents to Gold and Silver. These games follow the story of GS exactly, but the fact that it's a game with 2 regions is really enticing. The legendaries are elaborated on, the Safari Zone was put back an upgraded, we had Pokeathlons, the Battle Frontier and, best of all, Pokémon following you in the overworld. This feature, the time and effort that went into making all these overworld sprites for the 493 Pokémon at the time; it was unheard of, and amazing. The journey really felt personal, and you bonded with your team like never before. This is why HGSS is my personal favourite main-series set of the whole franchise, and why it falls under the second category; above and beyond.

Finally, if we look at Omega Ruby and Alpha Sapphire, the 3DS equivalents to Ruby and Sapphire, we can see they've been presented in the same way FRLG were. They're just re-skins of RS with an emphasis on the Generation 6 feature of Mega Evolution, as well as a post story in the form of the Delta Episode. However, I feel that ORAS did remakes better than FRLG, but they definitely didn't live up to the standard that HGSS set for us. I feel like most people are a bit harsh on ORAS 7.8/10 for too much water in a game where water is half the theme? Seriously? but I can defnintely see where they're coming from, and I do agree that it belongs in the frsh coat of paint category.