Neither scenario is correct. There is no opportunity to use the power declared in Turn A during Turn B because once that turn starts the chosen power has already expired. It would follow closer to this:

Turn A: At the beginning of your turn Nico is given whatever power which can take effect immediately (ie. Poison, Outwit) during the beginning of Turn A.

Turn B: As the turn begins whatever power you assigned Nico in Turn A ends. At the beginning of your turn you then have the option to declare another power so long as it is not the same one that was declared in Turn A.

Ok, one concern/question then:

Turn A: during the "Beginning of Turn" use staff to gain poison and use it.
Turn B: 2 actions to resolve during "Beginning of Turn": poison and the loss of the power. Use Poison again then lose it.

Am I seeing things here? My gut says this shouldn't work this way, but my head is reading that, yes it does.

I'm not sure where some folks above are at, I just wanted to add the following clarification:

Any power that Nico is using during turn A is gone before the beginning of turn B. With the start of turn B she can choose a different power, even during the beginning of turn and use that power (if it's appropriate to do so). But at no point in time will she ever have 2 powers nor will the effects of the power from the previous turn still be going during the following turn.

(Hoping I didn't just make it worse)

If you chose Poison on turn A, you cannot use it on Turn B. In fact, Nico would lose poison before the beginning of turn B.

just to put in out there to make sure its not lost in all the explanation- when you lose outwit to declare poison, the outwitted power is no longer countered.
(yes i know that she loses it before turn B, just wanted to make sure to note the loss of outwit un-counters the power)

Yeah, Turn A's actions end when your next turn begins. Simple enough.