n-category = (n,n)-category
n-groupoid = (n,0)-category
A 2-poset is any of several concepts that generalize posets one step in higher category theory. One does not usually hear about -posets by themselves but instead as special cases of -categories, such as the locally posetal ones.
Fix a meaning of -category, however weak or strict you wish. Then a -poset is an -category such that all parallel pairs of -morphisms are equivalent for . Thus, up to equivalence, there is no point in mentioning anything beyond -morphisms, not even whether two given parallel -morphisms are equivalent. This definition may give a concept more general than a locally posetal -category for your preferred definition of -category, but it will be equivalent if you ignore irrelevant data.