That's not really the same sort of "probabilistic notion of primality". This isn't about a test for primality; this is about a probabilistic model of the prime

That's not really the same sort of "probabilistic notion of primality". This isn't about a test for primality; this is about a probabilistic model of the primes, where we replace the actual primes by a randomly-generated subset of the natural numbers that we believe ought to have similar properties. The basic one is the Cramér model, where we say that n is prime with probability 1/(log n), and these are independent of one another. This is giving a more complicated such model which it claims agrees better with the statistics of the actual primes.