I remember this chestnut. Taken together, your two equations say that the torus $i(T)$ is contained in a 3-ball $B$ embedded in $S^2 \times S^1$: the first equation says that $i(T)$ is obtained up to isotopy by drilling a hole through $B$ and taking the resulting boundary; the second equation says that $i(T)$ is the boundary of a thickened knot in $B$ (at least, I think that's what your notation means...). However, these are not the only possibilities. There are separating embeddings of $T$ in $S^2 \times S^1$ which are trivial on $\pi_1$ but are not contained in a ball: take the boundary of Whitehead's link in $S^1 \times D^2$ (described in Rolfsen) and then embed $S^1 \times D^2 \to S^1 \times S^2$ in the standard way.

I remember this chestnut. Taken together, your two equations say that the torus $i(T)$ is contained in a 3-ball $B$ embedded in $S^2 \times S^1$: the first equation says that $i(T)$ is obtained up to isotopy by drilling a knotted hole through $B$ and taking the resulting boundary; the second equation says that $i(T)$ is the boundary of a thickened knot in $B$ (at least, I think that's what your notation means...). However, these are not the only possibilities. There are separating embeddings of $T$ in $S^2 \times S^1$ which are trivial on $\pi_1$ but are not contained in a ball: take the boundary of Whitehead's link in $S^1 \times D^2$ (described in Rolfsen) and then embed $S^1 \times D^2 \to S^1 \times S^2$ in the standard way.