Are there any examples of rings $R$ such that $\mathrm{End}(R,+,0)\not\cong R$?

The correct statement is that a ring $R$ is precisely the endomorphism ring of $R$ as a right $R$-module ("Cayley's theorem for rings"). When $R$ is any quotient of $\mathbb{Z}$, a morphism of right $R$-modules is just a morphism of abelian groups, which gives the examples you describe.

In general, preserving right $R$-module structure is stronger. For example, if $R = \mathbb{Z} \times \mathbb{Z}$ then the endomorphisms of the additive group of $R$ are given by $2 \times 2$ integer matrices. For a field counterexample, if $R = \mathbb{R}$ then $R$, as an abelian group, is a vector space over $\mathbb{Q}$ of uncountable dimension and therefore it admits an enormous endomorphism ring. (For a simpler field counterexample, if $R = \mathbb{Q}(i)$ then $R$, as an abelian group, is $\mathbb{Q}^2$, so its endomorphism ring is $2 \times 2$ rational matrices.)