Traveling-wave convection patterns in binary fluid mixtures of ethanol and water have been studied experimentally in containers with lateral boundaries of different shapes, including circular cells of two different sizes, rectangular and stadium-shaped cells. The nature of the patterns depends qualitatively upon the shape of the lateral boundaries. This paper analyzes these patterns in terms of topological phase defects in the complex order parameter, providing a reduced description of the system. The experimentally measured phase fields (and hence the patterns) can be reconstructed from the location and charge of the topological defects. For good agreement with experiment, the structure of individual defects is found to depend on both nearest-neighbor interactions of like-sign defects and the global structure of the patterns, reflected by the net charge of the defects. Further implications of these results for understanding traveling-wave patterns and dynamics are discussed.