  We extend results of Br\"ustle-Yang on ideal quotients of 2-term
subcategories of perfect derived categories of non-positive dg algebras to a
relative setting. We find a new interpretation of such quotients: they appear
as prototypical examples of a new construction of quotients of extriangulated
categories by ideals generated by morphisms from injectives to projectives. We
apply our results to Frobenius exact cluster categories and Higgs categories
with suitable relative extriangulated structures, and to categories of walks
related to gentle algebras. In all three cases, the extriangulated structures
are well-behaved (they are 0-Auslander) and their quotients are equivalent to
homotopy categories of two-term complexes of projectives over suitable
finite-dimensional algebras.
