The distinction is between independent identically distributed random variables, where it's possible to not get the item after 1/p tries, and what you described, which the events are not identically distributed. In the case you describe, the probability you'll get an item when you have a second completion is 1/9, not 1/10.

This is the distinction that the original article highlighted and gives rise to the Poisson distribution, which can have unintuitive results because it is not a uniform distribution. You are describing a uniform distribution. (A Poisson distribution characterizes the number of hits you would expect after running some number of iid random events.)