Are these two definitions of a semimodule basis equivalent?

I think we gave up on the "normal" definition of a quotient semimodule too early, and then got distracted with the Bourne relation.

We can't write $r\sim s\iff r-s\in N$ as you said, but we can write $r\sim s \iff r+N=s+N$.

This resembles the Bourne relation written above, but of course the quantifiers are different. This equality of sets allows much more flexibility, and not just the existence of two things in $N$ "lining up" $r$ and $s$.

Now if $M/N=0$ in this relation, and $M$ and $N$ contain additive identity $0$ (which I presume was intended to be kept in the definition of semimodule above) then $m\sim 0$ implies $m+N=0+N$ for all $m\in M$, and so in particular $m+0\in N$, i.e. $m\in N$. Hence, $M\subseteq N$.

So I think your original argument does work.