The definition of henselian is: A valued field $(\mathbb{K},\nu)$ is said to be Henselian if for any algebraic extension $\mathbb{L}$ of $\mathbb{K}$ there is a unique valuation $\tilde{\nu}$ on $\mathbb{L}$ such that $\mathcal{O}_{\mathbb{L}} \cap \mathbb{K} = \mathcal{O}_{\mathbb{K}}$.
But isn't it always true for any valuation v on an algebraic extension L of K that $\mathcal{O}_{\mathbb{L}} \cap \mathbb{K} = \mathcal{O}_{\mathbb{K}}$ ?
If that is the case, would it mean that there is only one possible valuation on an algebraic extension L of an henselian field K?