Summary

Using Generalized Linear Models, you have fit three different models for count data. The negative binomial regression was shown not to offer any improvement over the Poisson regression. The overdispersed Poisson regression seems to offer a reasonable alternative to the standard Poisson model, but there is not a formal test for choosing between them.

As a further avenue of exploration this dataset, consider that the existing model assumes, for a given ship type, that the damage incident rate is independent across different periods of construction and operation. In fact, you may very reasonably assume that these observations are correlated, in which case fitting a model using the Generalized Estimating Equations procedure would be appropriate.