Hi Andrew,
Yeah it makes sense to me that the second model works better: I think there is a lot of over-dispersion in your data, which is very common in Poisson models – counts are very often over-dispersed relative to Poisson expectations.
What your second model does is basically allowing each observation to have its own Poisson rate, which enables to get more of the variation in the data. This is the idea behind gamma-Poisson (aka negative-binomial) models – look at paragraphs 11H1 and 11H2 of this NB for a detailed explanation.
Even better would be to use a hierarchical structure: each cluster would have its own expected Poisson rate, and this rate would both inform and be informed by the higher-level parameters of the population (i.e the parameters across clusters).
Hope this helps 