Here's an alternative solution. We have that $X$ is a torus unioned with a disk which bisects the torus. Let $X_1 \subset X$ be the space containing the disk, two small strips on the torus where the torus and disk intersect, and a small strip over the top of the torus connecting the inside and outside circles (see my diagram below).
X_1">
Let $X_2 \subset X$ be the torus along with small strips near were the torus and disk intersect (again, see below.)
X_2">
Then, $X_1$ and $X_2$ are open, $X_1 \cup X_2 = X$ and $X_1 \cap X_2$ is path connected, so w emay apply Seifert van Kampen. We have that $X_1$ retracts to a circle, so $\pi_1(X_1) = \langle \alpha \rangle$. We can deformation retract $X_2$ to the torus, so $\pi_1(X_2) = \langle \beta, \gamma\rangle ~\mid~\beta\gamma\overline{\beta}\overline{\gamma}\rangle$. Finally, $X_1 \cap X_2 \sim S^1 \wedge S^1$, so $\pi_1(X_1 \cap X_2) = \langle \delta, \epsilon\rangle$. All these loops are shown on $X$ below.
X">
From the diagram, we find that $\delta$ and $\epsilon$ are null homotopic in $X_1$ and that $\delta \sim \epsilon \sim \beta$ in $X_2$, so by SvK the fundamental group of $X$ is $\pi_1(X) = \langle \alpha, \beta, \gamma ~\mid \beta\gamma\overline{\beta}\overline{\gamma}, \beta\rangle \cong \langle \alpha, \gamma\rangle.$