Why can't I transform a distribution by deducting one from all counts?

Suppose I have records of the number of fishes that each fisherman caught from a particular lake within the year. The distribution peaks at count = 1 (i.e. most fishermen caught just one fish from the lake in the year), tapers off after that, and has a long right-tail (a very small number of fishermen caught over 100 fishes).

Such a data could possibly fit either a Poisson Distribution or a Negative Binomial Distribution. However, both of these distributions have …

count datascience distribution fish lake records