Q:

How does Thief's Poisoner move work? 

Quoting from the official playbook, Poisoner says:

You’ve mastered the care and use of a poison.  Choose a poison from the list below; that poison is no longer dangerous for you to use.  You also start with three uses of the poison you choose.  Whenever you have time to gather materials and a safe place to brew you can make three uses of the poison you choose for free.  Note that some poisons are applied, meaning you have to carefully apply it to the target or something they eat or drink.  Touch poisons just need to touch the target, they can even be used on the blade of a weapon. 

I understand that you can use and craft without any risks one poison of your choice, the one you've mastered.  However the description says: ...you can make three uses of the poison you choose... which leads to confusion.  Does this really mean the poison you chose/the poison you've chosen? 
Would this mean you won't be able to craft any other poison but the one you chose? 
Or is perhaps Poison Master allowing you to master new poisons, making them no longer dangerous to use AND ALSO craftable? 

A:

Everywhere that it says "the poison you choose", it means, the one that you choose when you take the move.   From a very literal reading, you could say that each time this phrase occurs it's a separate choice, but I would not interpret it that way.   (It would be clearer if they simply said "that poison", to obviously refer back to the first "choose as poison".) 
Glazius correctly points out that another move, Brewer, exists specifically to allow you to learn to craft other poisons.   So:

Poisoner lets you craft and safely use one poison
Poison Master lets you safely use any other poison - after you've used it once
Brewer lets you craft any one other poison that you've used at least once. 
Alchemist replaces Brewer, and lets you craft any poison that you've used at least once, and also invent new poisons

It also doesn't say you can't try to use or craft any poison - just that it's not safe.   In fact you must use other poisons in order to learn them.   Roll Defy Danger! 

A:

You're thinking of "Brewer". 

When you have time to gather materials and a safe place to brew you can create three doses of any one poison you've used before. 
-- "The Thief", via the Github. 

If you were meant to be able to create three doses of any poison from the start, this move would be useless to you.  So it should be interpreted that "Poisoner" only lets you create the poison you chose to master during character creation. 
It should be noted that by itself "Brewer" doesn't make poisons safe to use; you'll still want "Poison Master" for that.