Let $s_0>0$. The right statement is that the following are equivalent:

• The sum $\sum_{n=1} \tfrac{\mu(n)}{n^s}$ converges for $s>s_0$.

• $\zeta(s)$ has no zeroes with real part $>s_0$.

• $1/\zeta(s)$ has an analytic extension to the half space $\mathrm{Re}(s) > s_0$

• $\sum_{n=1}^N \mu(n) = O(N^{s_0+\epsilon})$ for all $\epsilon >0$.

As you point out, $1/\zeta(s)$ certainly has an analytic continuation to the real interval $(0,\infty)$, since $\zeta$ doesn't vanish on the positive reals.

Let $s_0>0$. The right statement is that the following are equivalent:

• The sum $\sum_{n=1} \tfrac{\mu(n)}{n^s}$ converges for $s>s_0$.

• $\zeta(s)$ has no zeroes with real part $>s_0$.

• $1/\zeta(s)$ has an analytic continuation to the half space $\mathrm{Re}(s) > s_0$

• $\sum_{n=1}^N \mu(n) = O(N^{s_0+\epsilon})$ for all $\epsilon >0$.

As you point out, $1/\zeta(s)$ certainly has an analytic continuation to the real interval $(0,\infty)$, since $\zeta$ doesn't vanish on the positive reals.