Now I'm looking at the following generalisation of $\kappa$-closedness of a filter on a set to a filter in a partial order:
Let $\kappa$ be a regular uncountable cardinal. A partial order $P$ is $\kappa$-closed if every decreasing sequence $\langle p_\xi : \xi < \lambda \rangle$ of length $\lambda < \kappa$ has a lower bound in $P$.
The "decreasing" seems to restrict the definition to considering totally ordered subset only. Is this really correct? Or should it perhaps be arbitrary subsets of size less than $\kappa$ (in analogy to the definition for filters on sets)?
No, the definition is correct. What you think of is $\kappa$-directed (or $<\kappa$-directed in some terminology) which state that every subset of size $<\kappa$ has a lower bound.
This is a strictly stronger definition than $\kappa$-closed.