In standard topological terms, the exact sequence that relates homotopy groups of the base $B$, fiber $F$ and total space $E$ of topological fibration gives

$$\pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \pi_0(F),$$

that is,

$$ 0\to \pi_1(T-4) \to \pi_1(S^2-4) \to \mathbb Z_2.$$

The middle map works by taking a loop above and pushing it to the base; the right map works by taking a loop on $S^2-4$, lifting it *as a path*, and taking 0 or 1 depending on whether the resulting path is closed or not.

I believe the OP described the fundamental group of the torus as $\left< a, b, c, d, e, f| [a, b]cdef = 1\right>$ where $a, b$ are two circles of the torus and $c, d, e, f$ are four loops around the holes. For $S^2-4$ my suggestion would be to use $\left< C, D, E, F| CDEF = 1\right> = \left< C, D, E\right>$ where $c$ is over $C$ etc (rather then $g, h, x, w$).

Now, since the loop around $c$ on torus has to wind twice around it when projected to the sphere (think about complex $z\mapsto z^2$ map) it's easy to see that $c\mapsto C^2$, $d\mapsto D^2$, $e\mapsto E^2$, $f\mapsto F^2$.

What about $a$ and $b$? Careful observer should note that it's a bit tricky to define the loops. E.g, if you move $a$ parallel to itself, you'll get a new $a'$ which would differ by something like $cd$ depending on which points are where and depending on how you draw the basepoints on the loops.

For the exact calculations one should fix the torus to be $\mathbb R\times \mathbb R/\mathbb Z\times\mathbb Z$ so that fixed points of $z\mapsto z$ are the vertices $c = (0, 0)$, $d = (1/2, 0)$, $e = (0, 1/2)$, $f = (1/2, 1/2)$. Moreover, you should now select some basepoint and draw the cycles around $c, d, e, f$ so that $cdef = 1 $.

Unfortunately, from the picture it's not easy to say where some simple loops like horizontal or vertical go. While in homology they seem to be $C+D$ and $D+F$, one has to draw them really carefully with the basepoints. I couldn't do that, but here's something different instead.

I tried to exhibit some expressions $a, b$, which may be not exactly the cycles above, but which nevertheless satisfy $[a, b]cdef \mapsto 1$. In other words, these $a, b$ will be generators, but different ones.

I was able to make $a = CDEC^{-1}, b = CCDC^{-1}$ work:

$$CDEC^{-1}CCDC^{-1}(CDEC^{-1})^{-1}(CCDC^{-1})^{-1}CCDDEE(CDECDE)^{-1} = $$

$$ = CDEC^{-1}CCDC^{-1}CE^{-1}D^{-1}C^{-1}CD^{-1}C^{-1}C^{-1}CCDDEE(CDECDE)^{-1} = $$

$$ = CDEC^{-1}CCDE^{-1}D^{-1}D^{-1}DDEE(CDECDE)^{-1} = $$

$$ = CDECDE (CDECDE)^{-1} = 1 $$

I think this is more-or-less the explicit map you're asking for!

Finally, note that the map $ \left< C, D, E, F| CDEF = 1\right>
\to \mathbb Z_2$ is given by counting all the letters modulo 2 (consistent because $F = 1 = 3 = (CDE)^{-1}$), so the image of the map discussed above should contain exactly expressions with even number of letters.