We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting object, the $(d+2)$-angulated cluster category, and the $d$-almost positive subcategory of the derived category (the higher analogue of the category of two-term complexes of projectives). We show that the third one, introduced by the second-named author, is the $d$-exangulated quotient of the other two, introduced by Oppermann and Thomas, by the ideals generated by morphisms factoring through morphisms from injective to projective objects, thus providing an algebraic connection between the two models of Oppermann-Thomas. This is a $d$-exangulated version in type $A$ of a result of Brüstle and Yang and its interpretation by the first-named author together with Fang, Palu, Plamondon and Pressland. It also explains a well-known coincidence between the number of 2-term silting complexes in type $A_{n}$ and of tilting modules in type $A_{n+1}$ from the $0$-Auslander perspective. We expect this to serve as a prototypical example of the $0$-Auslander correspondence in higher homological algebra.