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Seifert fiberable manifolds with several Seifert fiberings

I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds: Hatcher: Notes on Basic 3-Manifold Topology.

Regarding the class d), it follows from Proposition 2.1 on page 31, that $M(0,0;1/2,-1/2,\alpha/\beta)$ is fiber-diffeomorphic to $M(0,0;1/2,-1/2,\alpha'/\beta')$ if and only if $\alpha/\beta=\pm \alpha'/\beta'$ if and only if $M(-1,0;\beta/\alpha)$ is fiber-diffeomorphic to $M(-1,0;\beta'/\alpha')$. But it is not clear to me, that for different $q,q'\in\mathbb{Q}$ with $q,q'\geq 0$ the manifolds $M(-1,0;q)$ and $M(-1,0;q')$ are not diffeomorphic and also not diffeomorphic to the manifolds under a),b),c),e), i.e. not diffeomorphic to the solid torus, the twisted $I$-bundle or the twisted $S_1$-bundle over the Klein bottle, or any lens space.

Is this true anyway? And do those manifolds under d) have a name (i.e. are known under a certain name like lens spaces)?