There are a host of open source remakes of old game engines that fix bugs and update them for modern systems as well as add support for higher resolutions and widescreen aspect ratios. Here's a few off the top of my head:

• OpenMW for The Elder Scrolls III: Morrowind. This is probably the most famous one.
• Daggerfall Unity for The Elder Scrolls II: Daggerfall.
• OpenRA for Red Alert, Tiberian Dawn, Dune 2000, and (in pre-alpha form) Tiberian Sun and Red Alert 2 (the latter requiring extra fiddling with Github repositories).
• KeeperFX for Dungeon Keeper.
• OpenRCT2 for Roller Coaster Tycoon 2.
• OpenTTD for Transport Tycoon Deluxe.
• OpenJKDF2 for Dark Forces 2 and its standalone Mysteries of the Sith expansion.
• Arx Libertatis for Arx Fatalis, though IIRC the version sold now integrates an older build of it; one without the fixes for glyph drawing - it's worth upgrading for that alone as spellcasting is a nightmare without the fix.
• TRX for Tomb Raider I and II.
• Tfix, T2Fix, and the Sneaky Upgrade for Thief 1/2/3 respectively. These are mods, not wholesale engine replacements, but serve the same purpose.

These are just the ones I know of. There are probably loads more.

Edit: the Ur-Quan Masters for Star Control II. I can't believe I forgot about this! Star Control II is one of the best and most influential games that most people have never heard about.

Viva New Vegas for Fallout: New Vegas and the Unofficial Patch for VtM: Bloodlines are my go-to examples. Not a mod but Ninja Gaiden Black is the definitive version and much better than both Ninja Gaiden (2004) and Ninja Gaiden Sigma.

For League of Legends the definitive way to play is to not start. I swear I do not know of any other game that has such addictive properties while being so absolutely fucking awful. You don’t stop playing this game, You take extended breaks.
Friends don’t let friends play League

First off, here's a big list of unofficial ports:

https://www.pcgamingwiki.com/wiki/List_of_unofficial_ports

Dhewm 3 for Doom 3. Just updates the game engine to run better on modern hardware since id tech 4 is open source. Mod support sort of, not every mod is compatible. They helpfully list those that do here: https://dhewm3.org/mods.html

The Sims 2 High DPI patch, self explanatory:
https://github.com/lah7/sims2-4k-ui-patch

VCMI - Heroes of Might and Magic 3 open source engine/game extension. Has a built in mod manager, really handy: https://vcmi.eu/

Coming soon / WIP: I'm excited to play this port of the MS-DOS game Albion when it reaches feature parity with the original:

https://github.com/csinkers/ualbion

For Chrono Trigger, definitely don't play the original SNES localization. No disrespect to Ted Woolsey, he was one man working on an unreasonably tight deadline and hard technical limitations, but the retranslation is much better.

Pokemon Red at 10fps on a gateway laptop circa 2004

The definitive way to play Skyrim is to take a shot every time someone says "dragon".

I've been playing BG3 with a loot randomizer. It really makes you have to think about your builds because you can't rely on certain gear anymore. Makes it almost feel like a roguelike.

Silent Hill - alone, in the dark.

I have been thinking of whether there was a pokemon mod that does something like this.

I’d like something that gets rid of trade evolutions and puts all 151 in a single game.

I’m sure it’s out there. So, I guess that’s my half assed attempt at an answer.

Factorio. It doesn't matter how you are playing it, you are doing something wrong and that invalidates your whole base.

if you're playing FFXIV, especially on Linux, don't bother with the official launcher/updater. Use XIVLauncher, hell even use that if you're playing on Windows. Updates a hell of a lot faster plus you also get the dalamud stuff which adds a few quality of life plugin improvements.

In the time since Quake released, common rendering systems and resolution options on monitors have changed. ID's solution to put it back on Steam was some gargantuan monolith wrapper that might've used Unity or something, and ties to an online ID, so that it could release on consoles. The open source community's solution was to take the original, open-source engine release, and port it upwards. Playing through the recent Quake Brutalist Jam 3, a map pack using a set of reinvented weapons and altered enemies, they recommend you use the "ironwail" source port, which even has a native Linux build.

Huh, haven't heard of Bezelbub mod, I'll have to check if its compatible with DevilutionX the open source reimplementation of Diablo 1. I played DevilutionX on my steamdeck, but got stuck in beginning of nightmare as my character wasn't powerful enough to continue.

Fallout Fixt for anyone wanting to dive into the OG fallout.

With headphones.

The first Kingdom Come: Deliverance was a great game and I think it should be played first, but I understand it's old clunky and unoptimized. These days it should be played with certain QoL mods like unlimited saving, instant herb picking (no animation), richer merchants, and trough washing deluxe/clean Henry (honestly I don't want to spend all my time going to the baths or looking shite). There are others, probably about 7-8 total, that I recommend but you get the idea.

There's something like 20 versions of Machaarior 2. Dos versions are the best. 

https://mech2.org/forum/viewtopic.php?t=281

Quite niche, but sim racing, I will pick Assetto Corsa Rally as an example.

The definitive way to play is with a proper rig. 9nm+ Direct Drive wheel base, round wheel, a sequential shifter, handbrake and decent load cell pedals. Put it up on a nice ultrawide or triple screen and have an amazing time sliding through the forest. You can also do VR, for complete immersion as well.

Buuut that's also the expensive way, you can also have a load of fun with a controller or cheaper wheel on a desk.