Let $R$ be a ring with unity and $M$ a left $R$-module. Define the annihilator of $r\in R$ with respect to $M$ as $$(0:_Mr)=\{m\in M:rm=0\}.~~~~~~~~~~~~(1)$$ Similarly, for any submodule $N$ of $M$, we can define the annihilator of $r$ with respect to quotient $M/N$ as $$(0:_{M/N}r):=\{m+N\in M/N:r(m+N)\in N\}=\{m\in M:rm\in N\}=(N:_{M}r).~~~~(2)$$
My questions:
Are there examples of modules or ring element(s) or conditions for which property (2) holds? In particular, for a case when $rm\neq 0$.
What you know is $m+N\in (0:_{M/N} r)\iff rm\in N\iff m\in(N:_M r)$, which does not mean the same thing as the equality you wrote.
It would be correct to write $(N:_M r)/N=(0:_{M/N} r)$, though.
As far as I can see that holds for all rings and modules.
As for a nontrivial case, why not keep things simple and just try $R=\mathbb Z$ and $M=\mathbb Z/4\mathbb Z$, $N=2\mathbb Z/4\mathbb Z$.
With $r=2$ you have, for example,
$$ (N:_M r)=M $$
$$ (0:_{M/N} r)= M/N $$
and $m=1 +4\mathbb Z$ is nonzero such that $rm\neq 0+4\mathbb Z$.