Another way to see that there are four homotopy classes of maps $S^1\times S^1 \to SO(3)$ is to use the fact that $SO(3) = \mathbb{RP}^3$. So by cellular approximation,
\begin{align*} [S^1\times S^1, SO(3)] &= [S^1\times S^1, \mathbb{RP}^3]\\ &= [S^1\times S^1, \mathbb{RP}^{\infty}]\\ &= [S^1\times S^1, K(\mathbb{Z}_2, 1)]\\ &= H^1(S^1\times S^1; \mathbb{Z}_2)\\ &\cong \mathbb{Z}_2^2. \end{align*}
More generally, the same argument shows that the set of homotopy classes of maps $\Sigma_g \to SO(3)$ is in bijective correspondence with $H^1(\Sigma_g; \mathbb{Z}_2)$. In particular, there are $2^{2g}$ such classes, which is consistent with the statement in the final paragraph of Dan Ramras' answer.