I have often heard people talk about, say, "the" twisted $S^2$bundle over $S^2$. My question is, what do they mean by a twisted bundle? I know that in the above example any $S^2$bundle over $S^2$ is either $S^2 \times S^2$ or a unique nontrivial $S^2$bundle over $S^2$. But is that how it is defined? Is a twisted bundle just a nontrivial one, or is it a specific bundle among the nontrivial ones?
$S^2$ bundles over $S^2$ (smooth, PL, topological) are in bijective correspondence with homotopyclasses of maps: $$ S^2 \to BSO_3 $$ which up to homotopy is $$\pi_2 BSO_3 \simeq \pi_1 SO_3 \simeq \mathbb Z_2 $$ So there's precisely two nonisomorphic $S^2$bundles over $S^2$. You can view the nontrivial $S^2$bundle as the fibrewise onepoint compactification of the vector bundle over $S^2$ whose Euler class is $1$. That vector bundle has fairly standard constructions. 

