One possible motivation for considering the triangles in 1. is that they induce long exact sequences
$\cdots \rightarrow Hom_K(Z,C_f[-1]) \rightarrow Hom_K(Z,X) \rightarrow Hom_K(Z,Y)\rightarrow Hom_K(Z,C_f) \rightarrow \cdots$
and
$\cdots \rightarrow Hom_K(C_f,Z) \rightarrow Hom_K(Y,Z)\rightarrow Hom_K(X,Z) \rightarrow Hom_K(C_f[-1],Z) \rightarrow \cdots$
for any $Z$.
I think that the triangulated structure of $K(\mathcal{A})$ reflects (in some sense "up to homotopy") the abelian structure of $C(\mathcal{A})$. Indeed $K(\mathcal{A})$ is the stable category (see the book of D. Happel "Triangulated categories in the representation theory of finite dimensional algebras") associated with the abelian category $C(\mathcal{A})$.
