First I would like to be clear about the definition, which I am having trouble finding.

What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion of $A$ at its maximal ideal).

1- Does it mean that every element of $B$ satisfying a polynomial with coefficients in $A$, belongs to $A$?

2- Does it mean the same as above but assuming $b$ is an element of some commutative finite extension of $B$?

Afterwards I am trying to understand why if $A$ is local, noetherian, Henselian, and $\hat{A}$ is normal, characteristic is zero, then $A$ is algebraically closed in $\hat{A}$.