I have a problem with understanding what is a neighbourhood of a singular fiber in a Seifert fibered space coming from the zero surgery. For me a 3-manifold $Y$ is a SFS if it has a decomposition into a disjoint sum of circles s.t. every circle has a neighbourhood of form $V(p,q)$, where $V(p,q)$ is obtained from $D^2 \times I$ by idenitying the top and the bottom disk by a rotation of angle $\frac{2\pi q}{p}$. Circles for which the neighbourhood is equal to $V(1, 0)$ are called regular fibers and others are called singular fibers.

Let's say that I have a Seifert fiber structure on an oriented three-manifold $Y$ with a Seifert invariant given by $\{b,g; \frac{a_1}{b_1}, \ldots, \frac{a_n}{b_n}\}$.

If I perform the zero surgery on a regular fiber of $Y$, then what is the Seifert fiber structure on the resulting space?

For a $\frac{p}{q}$-surgery it seems that the invariant should be given by $\{b,g; \frac{a_1}{b_1}, \ldots, \frac{a_n}{b_n}, \frac{p}{q}\}$ as the neighbourhood of the newly created singular fiber is isomorphic to $V(p,q)$, however for the zero surgery case $V(0,1)$ doesn't make any sense to me. My geometric understanding of the situation hints me that we should adapt the convention that $V(0,1)$ means the sum of concentric circles in each slice $(D^2 \setminus \{0\}) \times \{z\} \subset D^2 \times S^1$ and the core cicle $\{0\} \times S^1$. However, no literature I've found seem to corroborate that.

Let's say that I want to understand Seifert fiber structure on $0$-surgeries on torus knots.

Is there any canonical Seifert structure on these spaces somewhere in the literature?

The way I'd create a Seifert fibration on $S^3_0(T(p,q))$ is to start with $S^3$ and fiber it with $T(p,q)$'s in the standard way. That way I obtain a Seifert structure with two singular fibers with neighbourhoods $V(p,q), V(q,p)$. Then performing the $0$-surgery on any regular fiber should give us $S^3_0(T(p,q))$ with an invariant $\{b,0; \frac{p}{q}, \frac{q}{p}, \frac{0}{1}\}$. This doesn't seem to be correct though (for example the fundamental group of this presentation would be $\mathbb{Z}_p * \mathbb{Z}_q$ - see my comments under ThiKu's answer).

Edit: as it was pointed out by Bruno Martelli in his answer, I confused here accidentally the actual zero-surgery on a knot with so called fibre-parallel surgery. The subject is discussed in depth in his excellent book "An Introduction to Geometric Topology" in chapter 10.3.13.