Is the more accurate statement "The Poisson distribution is the limit of the Binomial as n goes to infinity" not a better summary?

Maybe even better: it is the limit of the binomial distribution as the n goes to infinity with n * p constant, where p is the probability of success.

There is a dedicated wiki[1] page for relationship among probability distributions for anybody interested.

[1] https://en.wikipedia.org/wiki/Relationships_among_probabilit...