I am looking at the Mayer-Vietoris sequence for the suspension $\Sigma X$ of a space $X$, defined as $X\times [-1,1]$ with the usual identifications, with subsets $A, B$ defined to be the following $ A = X \times [-\tfrac{1}{2},1]/\sim, $ $ B = X \times [-1,\tfrac{1}{2}]/\sim, $ where $\sim$ is just the equivalence relation inherited from $\Sigma X$. Only the last part of Mayer-Vietoris is important to my question: $ 0 \to H_1(\Sigma X) \to H_0(X)\: \xrightarrow{f_*} \: H_0(A) \oplus H_0(B) \to H_0(\Sigma X) \to 0. $ Since $A$ and $B$ are contractible (they are homeomorphic to the cone on $X$, which is contractible), $H_0(A) \oplus H_0(B) \cong \mathbb{Z} \oplus \mathbb{Z}$.
Question: I am then given that $H_1(\Sigma X) \cong \ker f_* \cong \tilde{H}_0(X)$. The second isomorphism is the problem; I thought that the reduced $0$th homology was defined to be $\ker \big(g_*:H_0(X) \to H_0(pt)\big)$, where the codomain is the homology of a point. Isn't this different to what we have here?