I'm trying to figure out the difference between being $\kappa$-closed and strategically $\kappa$-closed (in the context of forcing).
A poset is $\kappa$-closed if every $\alpha$-chain with $\alpha<\kappa$ has a lower bound. A poset is strategically $\kappa$-closed if player I has a winning strategy in the game where both players successively choose an element below all previous moves, where player I moves first at limit steps and wins if the game lasts $\kappa$ rounds.
My initial question is to have an example of a strategically $\kappa$-closed poset that is not $\kappa$-closed.
My follow-up question would be if there is any significant difference in terms of forcing with a $\kappa$-closed poset instead of merely a strategically $\kappa$-closed poset.
Here's a silly example:
Let $\mathbb{P}$ be the poset consisting of all countable-ordinal-length sequences of countable ordinals $p$ satisfying $$p(\alpha)\le\alpha\mbox{ for every $\alpha\in\operatorname{dom}(p)$}$$ and $$\sup\{p(\alpha): \alpha<\lambda\}=\lambda \mbox{ for every limit }\lambda\in\operatorname{dom}(p),$$ ordered by reverse inclusion as usual. This is clearly not $\omega_1$-closed, but it is $\omega_1$-strategically closed.