For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like. For a partial order $\mathbb{P}$, the following are equivalent:

1) Forcing with $\mathbb{P}$ adds no new functions from ordinals to ordinals with domain $\kappa$.

2) The intersection of $\kappa$ many dense open subsets of $\mathbb{P}$ is dense.

$\kappa$-closure of partial orders is definitely not a forcing invariant, as witnessed just by the fact that an atomless complete boolean algebra is never countably closed. However, we can ask is there a characterization partial orders $\mathbb{P}$ in terms of what happens in their generic extensions that is equivalent to the following: The boolean completion of $\mathbb{P}$ has a $\kappa$-closed dense subset?

EDIT: Changed the question in light of the example mentioned by Dorais.