It seems to me that you are effectively asking why a $(0,1)$-category is a poset. Because if that is so, it makes sense to define an $n$-poset to be an $(n,n+1)$-category.

To see why a $(0,1)$-category is a poset, just unwind the definition: it contains possibly non-invertible 1-morphisms, but any two of them that have the same source and target are equivalent (and the space of choices of equivalences between them is contractible). By the characterization of posets as categories that means it is a poset.

It seems to me that you are effectively asking why a $(0,1)$-category is a poset. Because if that is so, it makes sense to define an $n$-poset to be an $(n-1,n)$-category.

To see why a $(0,1)$-category is a poset, just unwind the definition: it contains possibly non-invertible 1-morphisms, but any two of them that have the same source and target are equivalent (and the space of choices of equivalences between them is contractible). By the characterization of posets as categories that means it is a poset.