Traditionally, it is considered that data that fit a Poisson distribution best come from a Poisson process, e.g., a countable number of events during a certain time period. The example in the link you posted was a certain number of pieces of mail per day.
The Binomial distribution is the result of a Bernoulli process, e.g., the result of multiple trials with only two outcomes. A coin toss is the most boring but easy to understand example of a single Bernoulli trial. If you flipped the coin thousands of times you would approximate a Binomial distribution with a p=0.5, assuming your coin was fair (50/50 chance).
These two (the Poisson and Binomial) are indeed closely linked. Again, as your link pointed out, for relatively rare event Binomials, the Poisson distribution can substitute. Going back to the mail, if you considered every second or minute of the day a test of if you would get a piece of mail, then the vast majority of times you would fail this binomial test. (The p would be very low). You can instead approximate this with a Poisson. Of course, in this case, the Poisson is also more intuitive. A common use of the Poisson distribution is counting animals (in a forest, field, etc). You count a certain number of animals per area or time period. You could again, theoretically set up each minute as a test of if you'd find an animal or not (binomial), but no one would ever want to this.
So, aside from using my examples as a kind of heuristic, I guess it would be important to ask yourself:
1.) How common is a success?
2.) Is your time period continuous or discrete? This does the most to determine how "common" your success is, since when you coin flip you're not counting all the times you're not actively flipping as "failure". The clock starts and stops only when you're doing a single trial.