I'm reading the notes about Seiberg-Witten invariants by Salomon (see https://people.math.ethz.ch/~salamon/PREPRINTS/witsei.pdf). In Theorem 5.5 he gives an interesting classification of $\operatorname{spin}^c$ structures in terms of the characteristic line bundle associated to the structure. A consequence of this is that all the information about the $\operatorname{spin}^c$ structure can be recover from its characteristic line bundle iff there is no 2-torsion in the second integral cohomology group. However, in Exercise 5.9 he asks to prove that two $\operatorname{spin}^c$ structures are isomorphic iff the two characteristic line bundles are isomorphic. This seems to be in contradiction with the previous statement about the torsion because if there is a 2-torsion element in the second cohomology group then it is possible to create non isomorphic $\operatorname{spin}^c$ structures (using the twisted $\operatorname{spin}^c$ structure associated to the torsion element) with isomorphic characteristic line bundles.
What am I missing?