Specifying distribution in GBM In the GBM package, we can specify which distribution to use that represents our response variable.  I have count data and usually, we specify the distribution as Poisson for count data. But, when I check the distribution it is not Poisson distributed.
Is it okay if I specify the distribution in GBM as Poisson although the distribution is not Poisson distributed?

Yes, it is OK to use a Poisson family in a GBM even if the Poisson data are over-dispersed. Ultimately Poisson will be our loss function. What we care about is primarily our evaluation metric (MAE at first instance I suppose?) so it is down to the validation schema used and how the evaluation metric aligns to our use of the model. (CV.SE has a thread on "https://stats.stackexchange.com/questions/379264/" if one wants to explore this distinction further)
I would also note here that as $\lambda$ is quite low (~$0.05$), suggesting the potential for zero inflation, it make sense to consider a hurdle and/or a zero-inflated model too.
