There are several related notions dealing with similar notions. Historically the first is
Then Waldhausen came up with
with the applications in algebraic K-theory. His construction of K-theory for Quillen exact categories could be factored in two steps and the intermediate case is the Waldhausen category. From each Quillen exact category one can produce its stable category which is an example of a suspended category rather than triangulated. In particular, its suspension is not invertible in general it is in the case when the exact category is Frobenius (and in particular when it is abelian). A. Rosenberg has devised a generalization of suspended category to a nonadditive setting.
Finally there are more complete descriptions of stability in notions of stable model categories, studied mainly in 1990s, stable Segal categories of French school and stable quasi-categories of Joyal and of Lurie.
Should there be stable -categories at some point ?