n-category = (n,n)-category
n-groupoid = (n,0)-category
Fix a meaning of -category, however weak or strict you wish. Then a -category is an -category such that every 2-morphism is an equivalence and all parallel pairs of j-morphisms are equivalent for . Thus, up to equivalence, there is no point in mentioning anything beyond -morphisms, except whether two given parallel -morphisms are equivalent.
If you rephrase equivalence of -morphisms as equality, which gives the same result up to equivalence, then all that is left in this definition is a category. Thus one may also say that a -category is simply a category.
The point of all this is simply to fill in the general concept of -category; nobody thinks of -categories as a concept in their own right except simply as categories.