While it is technically possible to complete Myst without triggering them, these Achievements are expected to be completed when playing through Myst in the normal manner.
Consult a game walkthrough for more detail on completing these tasks.

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Description: Opened the safe in the cabin.

Description: Solved the gear puzzle inside of the clocktower.

Description: Raised the sunken ship.

Description: Viewed some dated constellations.

Description: Solved the generator power puzzle.

Description: Found a way back from Channelwood

Description: Found a way back from Mechanical.

Description: Solved the compass puzzle.

Description: Found a way back from Stoneship.

Description: Found a way back from Selenitic.

Description: Found the white page.

Description: Solved the fireplace puzzle.

These Achievements are obtained by triggering various endings to Myst.

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Description: Achieved the good ending.
Conditions: Link to D'ni while possessing the white page.

Description: Returned to Myst Island from D'ni.
Conditions: Achieve the good ending, then return using the Myst linking book.

Description: Achieved the good ending without inserting a single red or blue page.
Conditions: Most easily obtained using a Classic playthrough since puzzle solutions do not change, and all but guaranteed if attempting the Speedy Achievement. Start a new playthrough, and using knowledge from a completed game, obtain the white page from the docks, enter the fireplace code, and link to D'ni.

Description: Achieved the bad ending with Atrus.
Conditions: Link to D'ni without having obtained the white page.

Description: Achieved bad ending with Sirrus.
Conditions: Insert all 6 red pages into the red book, freeing Sirrus.

Description: Achieved bad ending with Achenar.
Conditions: Insert all 6 blue pages into the blue book, freeing Achenar.

Description: Achieved one of the brothers' endings while on a randomized playthrough.

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Description: Found Sirrus' message to Achenar.
Conditions: In Achenar's bedroom in Channelwood, push the far-right button on the holographic display device to view the message from Sirrus.

Description: Rode the tree to the highest position.
Conditions: Get in the tree elevator (treelevator?) near the cabin on the main island as it is rising and let it go all the way up. This may be most easily accomplished once you've reached the bottom. There is a wheel beside the elevator you can turn, then quickly get inside to begin riding it back up.

Description: Solved the piano puzzle in a single try.
Conditions: Have all the notes correctly set in the Rocketship the first time you slide the lever.
I recommend saving your game before sliding the lever to confirm your solution to the puzzle, so you can reload if your solution wasn't correct. Either carefully match the tones, or simply count by keeping in mind that the notches on the left and right of the sliders correspond to white and black keys on the keyboard, respectively, with the bottom corresponding to the left (low) side of the keyboard.

Description: Found the skeleton in the Mechanical Age.
Conditions: With the Fortress in the Mechanical Age rotated so that the bridge points to nothing, a skeleton can be seen through the telescope in Sirrus' bedroom. That's Bob!
More than one commenter has reported it is necessary to look for Bob before collecting the Mechanical Age's Red or Blue pages in order for the achievement to trigger.

Description: Made it through the maze in Selenitic without pressing "backtrack" on a randomized playthrough.
Conditions: The tones tell you which direction to go.
Unconfirmed, but it appears the tones may the same as in a Classic playthrough, and only the sequence of the path changes.
Ding: North
Rattle: West
Clonk: South
Pssht: East
Also, you may be able to backtrack without losing the Achievement by turning all the way around and only using the forward control, as long as you don't pull the joystick back into the "backtrack" position.

Description: Fill up all of the shelf space for a page type.
Conditions: Obtain all 6 pages for one of the brothers and place them all onto their shelf WITHOUT actually putting any of them into the corresponding book. On a Classic playthrough, you can collect all the pages from the Ages and use the known fireplace code to acquire the 6th. Alternatively, you can insert 5 pages into the book of one brother to learn the page number of the fireplace code without inserting any pages into the book of the other brother.

Description: Achieved the good ending within two minutes
NOTE: The Speedy Achievement was broken until Patch 8 (released October 7th, 2022) but should now be attainable.
Conditions: Start a New Game and select Classic, since the puzzle solutions do not change. You must: turn on all the marker switches, one of which requires setting the clocktower time to 2:40 to raise the path to its marker switch, then turn off the marker switch on the docks to obtain the white page. Then you must enter the fireplace, quickly enter the fireplace code, and link into D'ni using the green book. The timer starts the moment you gain control of your character on the dock. The Achievement should unlock as soon as you've linked into D'ni. It is possible to unlock the achievement using free movement in VR, provided you quickly activate running speed at the start.