Two extensions with the same Ext class are equivalent

Choose a projective presentation $0\to \mathrm{\Omega }M\stackrel{i}{\to }P\twoheadrightarrow M\to 0$. Projectivity lifts $P\to M$ through the middle epimorphism of each extension. Restricting those lifts to $\mathrm{\Omega }M$ gives maps $f,f^{\mathrm{\prime }}:\mathrm{\Omega }M\to N$. In the cokernel description of ${\mathrm{Ext}}^1(M,N)$, equality of the two classes says $f-f^{\mathrm{\prime }}=ti$ for some $t:P\to N$.

Each original extension is equivalent to the pushout of the projective presentation along its map $f$ or $f^{\mathrm{\prime }}$. The relation $f-f^{\mathrm{\prime }}=ti$ induces an isomorphism between these two pushouts, represented on $P\oplus N$ by $(p,n)\mapsto (p,n-t(p))$; it fixes both endpoint maps. Composing the endpoint- identity equivalences proves that $e$ and $e^{\mathrm{\prime }}$ are equivalent.