The binomial distribution can always be applied to model a binomial process, even with a large n and a small p. The normal approximation for a binomial distribution cannot, but that was not the question unless the product of j and o is sufficiently high.
More importantly: The Poisson distribution is not a special case of the binomial distribution. It is a different distribution used to model a different process. I’ll explain using meteors and meteorites.
The Poisson distribution models the number of events that occur in a given time period. For instance, the number of meteorite strikes on the Earth in a year. In this case, there is no limit to the number you are modeling (you can always have more meteorite strikes). The number you are modeling is also discrete (you can’t have 1.5 meteorite strikes).
The binomial distribution models the number of success in a sequence of independent events. For instance, the number of meteors that became meteorites. In this case, the limit to the number of meteorites is also the number of meteors. The number you’re modeling is discrete, like the Poisson distribution.
In the Poisson distribution, the “time interval” doesn’t come into play. It is the defining factor of the distribution.
Poisson and Binomial are two different distributions for two different processes. Understanding these differences fully requires a top-down fundamental understanding of both processes that can only be obtained through extensive reading and mathematical work.