When is a point process a Poisson point process?

Question

I've seen several equivalent definitions of Poisson point processes in the literature, one of which is: a random set $X \subset V$ satisfying:

1. Independence of counts of $X$ in disjoint subsets of $V$

2. Counts are Poisson-distributed

(See e.g. 2.3.1, Stochastic geometry and its applications.)

(2) is usually assumed, but are there alternate conditions in which the Poisson-distributed count emerges? (As this blog post seems to suggest, though the Poisson counts are perhaps assumed at some point.)

I.e., is there a sense in which the Poisson process is the canonical process with the independence property (1)? If so, any sources would be appreciated.