Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that $P$ is forcing equivalent to $\operatorname{Coll}(\kappa,\lambda)$? Or, more generally, which are the minimal conditions to make $P$ equivalent to $\operatorname{Coll}(\kappa,\lambda)$?

When $\lambda=\kappa$ (i.e., $\kappa$-Cohen forcing) I am pretty sure that this be true, so I was interested in understanding whether it generalizes.