= Solution
The curve $\mu$ crosses the alpha disk $\alpha_2$ once and is disjoint from $\alpha_1$, so in the presentation it represents $y$ up to inversion and conjugacy. The <Dehn filling> adds the relation $y=1$, after which the beta relator becomes $x^2$. Thus the filled genus-one Heegaard diagram has two alpha–beta intersections and
$$
\pi_1(M_{\mathcal H}(\mu))\cong\langle x\mid x^2\rangle\cong\mathbb Z/2.
$$
Compressing the displayed genus-two diagram along $\mu$ visibly cancels the annular handle and leaves that genus-one Heegaard diagram. It is therefore the lens space
$$
\boxed{M_{\mathcal H}(\mu)\cong L(2,1)\cong\mathbb{RP}^3.}
$$
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