You must choose to sneak instead of fighting your way through otherwise you'll miss the achievement. You need to avoid the guards or knock them out. This is done by using the Distract and the Knockout abilities. Beware they take time in order to be used again. Time your moves right and you should have no problems. This part is divided into several areas. If you are detected you teleport to the beginning of the area you are in. This does not void the achievement. The Cook does not void the achievement. You should get the achievement only when you complete the gate puzzle at the end (it's very simple).
So to clarify, I did it over again and didnt get caught (on my gamertag) and it still popped up. If you get caught and it resets you back to where you were before you were caught, dont worry, it will unlock as soon as you reach The Vault!!!!
Confirmed. If you agree with Tallis in the conversations about a stealthy approach, this cannot be missed and is not based on detection. I would bet if you tell her you'll do a full, frontal assault, you will miss this.