In hatcher (p.6) it says that $\mathbb RP^n$ can be obtained from $\mathbb RP^{n-1}$ by attaching an $n$-cell via the quotient map $S^{n-1} \to \mathbb RP^{n-1}$. I was wondering what this is for $n=0$: By definition $\mathbb RP^0 = \{\{-1,1\}\}$, which is homeomorphic to a one-point space. If we follow the described procedure we get an attaching map $p:S^0 \to \mathbb RP^0$. Clearly, $p$ is constant and hence we attach the interval $[0,1]$ to a one-point space, by identifying $0$ and $1$, which is $S^1$ and not $\mathbb RP^1 = S^1 / (x \sim -x)$.
Did I misinterprete what Hachter writes ?
You did not misinterpret Hatcher. $n=1$ is special: it's the only value of $n$ where $\Bbb{RP}^n \cong S^n$. As you noted, the two are homeomorphic in this case because they have identical cell structures.
For $n=0$ they're obviously distinct ($S^0$ is the two-point space, $\Bbb{RP}^0$ the one-point space), and for $n>1$ $\Bbb{RP}^n$ has fundamental group $\Bbb Z/2\Bbb Z$, while $S^n$ is simply connected. But for $n=1$ they're the same.