I'm trying to wrap my head around the "Clifford modules" definition of K-theory. Let's just deal with K-theory of a point. One common definition of the $-n^\text{th}$ K-group is the quotient$$K^{-n}=M(n)/i_n^*M(n+1),$$where $M(n)$ is the free abelian group of irreducible graded $C\ell(n)$ modules (ie the group of formal differences of modules) and $i_n^*:M(n+1)\rightarrow M(n)$ is the map induced by the inclusion of $C\ell(n)$ into $C\ell(n+1)$. (Alternatively, one can work with ungraded modules and shift the degree by one.) Meanwhile, it is also sometimes claimed (see Karoubi K-Theory III.4.15) that $K^{-n}$ is the subgroup of formal differences $[M,M']$ in $M(n)$ such that $i_{n-1}^*M=i_{n-1}^*M'$. Are these really the same?
For example, take $n=0$. The group $M(n)$ consists of formal differences of supervector spaces, and $i_{-1}$ is the map that forgets the grading. By the second definition, we're interested in pairs $[k^{i|j},k^{i'|j'}]$ with $i+j=i'+j'$, which are classified (up to equivalence) by $i+j\in\mathbb{Z}$. On the other hand, the $i_0$ map lands on pairs $[k^{i|i},k^{j|j}]$ (since these have an odd automorphism). Modding out $M(n)$ by these objects yields $K^0= (\mathbb{Z}\times\mathbb{Z})/\mathbb{Z}=\mathbb{Z}$, by the first definition. How are these two computations - one of $\mathbb{Z}$ as a subgroup, and one of $\mathbb{Z}$ as a quotient - related? Why should they agree for $n>0$?