Why is Poisson instead of Negative Binomial used for computing attacker's potential progress?
I have found the answer in this article by Meni Rosenfeld, page 7. The problem should indeed be modeled with a Negative Binomial distribution, and the computations done on Satoshi's paper are approximations.

The fact of matter is that the quality of the approximation does not rely on the expected high number of trials required to mine a block, but in the general similarity between the negative binomial and the Poisson distributions. This leads to non-negligible differences between the actual probability and the probability computed by Satoshi, e.g. for q=10% and 5 confirmations, Satoshi says P=17.73523%, while Rosenfeld says P=19.762%.