A proper model makes all the difference! The choice is at the discretion of the researcher but it should be guided by prior knowledge, theoretical foundations, objective comparison with alternative models, and evaluation of the model assumptions.
The parameter estimates are used in a linear combination to determine the distribution parameters. They can be difficult to interpret on their own. You might click the red triangle at the top and select Profilers > Profiler. You can change factor levels and see the change in the predicted response.
I have another idea about the lack of fit. I do not see the test for over-dispersion, which is a common occurance. The Poisson distribution has a single parameter. It is the mean and the variance of the distribution. Real distributions of counts often exhibit a variance that is greater than the mean. You should find check boxes when you select Generalized Linear Models for the over-dispersion tests and intervals and for the Firth bias-adjusted estimates. I recommend selecting both of these options.
Otherwise, you might consider adding terms for potential interaction and non-linear effects.
