Let $R$ be the ring $R=\mathbb C[x,y]/(x^2,xy,y^2)$, and let $B$ be the $5$-dimensional $R$-module with shape like a 'W'. That is, basis elements are $a_1,a_2,a_3,b_1,b_2$ and the module structure is given by $$y \cdot a_1=b_1,$$ $$x \cdot a_2=b_1,$$ $$y \cdot a_2=b_2,$$ $$x \cdot a_3=b_2,$$ and all other products of generators and basis elements are zero.

Let $A=\mathbb C$ and consider the two parallel morphisms $u,v \colon A \rightarrow B$ defined by $u(z)=zb_1$ and $v(z)=zb_2.$ Now ${\mathrm{coker}} \; u \simeq {\mathrm{coker}} \; v$ as $R$-modules, but $u$ and $v$ are non-isomorphic in $\mathrm{Mor}(\mathrm{Mod} \; R)$. This gives an example in the derived category of $\mathrm{Mod} \; R$.

Let $R$ be the ring $R=\mathbb C[x,y]$, and let $B$ be the $5$-dimensional $R$-module with shape like a 'W'. That is, basis elements are $a_1,a_2,a_3,b_1,b_2$ and the module structure is given by $$y \cdot a_1=b_1,$$ $$x \cdot a_2=b_1,$$ $$y \cdot a_2=b_2,$$ $$x \cdot a_3=b_2,$$ and all other products of generators and basis elements are zero.

Let $A=\mathbb C$ be the trivial $R$-module and consider the parallel morphisms $f,f' \colon A \rightarrow B$ defined by $f(z)=zb_1$ and $f'(z)=zb_2.$ Now ${\mathrm{coker}} \; f \simeq {\mathrm{coker}} \; f'$ as $R$-modules, but $f$ and $f'$ are non-isomorphic in $\mathrm{Mor}(\mathrm{Mod} \; R)$. This gives an example in the derived category of $\mathrm{Mod} \; R$.