Island Adventure

A Short Hike would probably be the closest thing I can think of to those criteria.

Spoiler
There is a second island - albeit quite small and not significant to the story.

Apart from that I'd say it meets all of the other criteria - you can explore the entire island without loading screens and you can reach the limits of the map so you can tell you're on an island.

Its mostly based on exploration but there are other activities you can find on the map. I'd say it meets the island theme pretty well also.

Oxenfree - Its horror/adventure that all takes place on one island.

Slime Rancher - Is more tropical and heavy emphasis on exploration.

Risen

Dead Island?

Tomb Raider (2013)

There's this shitty game if it counts, but can it really be called a game if you can do nothing?

Danganronpa 2 and Purrfect Date also take place on an island.

The Witness, Far Cry

Terraria
Epistory - Typing Chronicles
Batman Arkhan Asylum?
Monster Hunter: WORLD?
Tomb Raider
Your Island -KIMI NO SIMA-?
ISLAND
The Secret of Monkey Island?
Lost in Blue(Nintendo DS)

Gonna put the recommendations I made originally for reference:

• A Short Hike: Pretty much what you are describing.
• Minit: Which is sort of like Zelda
• Grow Home: Very vertical like A Short Hike, but rather than a world with people and puzzles, it's pretty empty and instead you create the world by growing vines upwards. There are a bunch of crystals to collect though, and I kind of love it's simplicity. One of my favorites.

Similar game idea, but not on island:

• Fausts Alptraum: Spooky Mansion instead (it's free too)
• The Norwood Suite : Weird Hotel instead
• Trick & Treat: Another Free spooky mansion to explore (more simplistic than Fausts Alptraum though)
• Future Proof: Takes place in a small world that about to end, but time always resets. Have to take advantage of not only the world, but time itself to save the world.

Multiple worlds/islands, but all follow the same idea:

• The Talos Principle: 3D puzzle game, that takes place in a set of small open worlds (some are even islands). The main puzzles take place in self contained sections, but there are sort of like meta puzzles in each world that requires more exploration. One of my favourite games.It's a bit more popular, so you might have heard of it.
• Jazzpunk: Has a couple of worlds. Pretty weird and more about finding all the dumb things you can do in the game. I would most compare it to The Norwood Suite.

Looking forward to what others recommend, and will try to think of more if I can.

Hudson's Adventure Island (NES)

Myst
Sleeping Dogs
Sinking Island
Return to Mysterious Island 1-2 (?)
GTA V
Test Drive Unlimited 1-2
Betrayer
Oxenfree
ArcaniA (meh)
The Deadly Tower of Monsters
I Am Alive (kinda of)
Heaven Island - VR MMO (haha)
N.E.R.O.: Nothing Ever Remains Obscure
Sir, You Are Being Hunted
How to Survive (?)
Psychonauts
Draw a Stickman: EPIC
TIMEframe (kinda of)
METAL GEAR SOLID V: GROUND ZEROES (or it is just a single base/camp?)

I enjoyed Miasmata a lot, but the game isn't for everyone. You are on a large island and your objective is to explore it to find different plants and make a cure for the plague. The game requires a ton of exploration and you have to use triangulation to known points to create a map or else you will get lost.

A Short Hike was a really good game that takes place on an island ans is all about exploration.

The Lost in Blue series on the DS was like a deserted island survival game. Anyone know if there is something like that on Steam?

Also the console version of The Sims: Castaways had an island-exploration and improvement story.

Castaway Paradise is a Steam game kind of like Animal Crossing.

Adventure Island 4 was tropical. took place on a single island, and (being the series's attempt a metroidvania) focused on exploring the island.

The first three games in the series probably wouldn't qualify, because they are broken into levels and the 2nd and 3rd explicitly feature multiple islands.

The original Far Cry also seems to fit your requirements perfectly.

Civilization on a pangea map