I'll ask this with a concrete example to be clear.
Let's say I have a Poisson process that tends to produce one event every two minutes. Then the probability of getting an event in a given minute is about 30%, and the probability of getting two events in a given minute is about 7.6%, assuming I'm applying the distribution correctly.
I'm interested in the time distribution between groups of events: How long do I have to wait to see two events within a minute? What is the general form for this?
I have been following this Question for about a day now, and am still not exactly sure what you are asking. Here are some comments based on sequences of one-minute time intervals, for which the number of $X$ of events in one minute is Poisson with mean 1/2.
EXACTLY ONE event in such an interval: $P(X = 1) = 0.3032653.$ Geometric mean waiting time for the first interval with exactly one event: reciprocal = 3.297443. By independence, given you have just seen such an interval, this is the waiting time for the next one. Negative binomial mean waiting time for two such intervals: double = 6.594885.
AT LEAST ONE event in such an interval: $P(X \ge 1) = 1 - P(X = 0) = 0.3934693.$ Geometric mean waiting time for the first interval with at least one event: 2.541494. Negative binomial mean waiting time for two such intervals: 5.082988. Perhaps coincidentally, this is the same as in the Comment where @Did speculates $E(D) = 1/[\lambda(1-e^{-\lambda})] = 5.082988,$ for $\lambda = 1/2.$
AT LEAST TWO events in such an interval: $P(X \ge 2) = 0.09020401.$ Geometric mean waiting time for the first interval with at least two events: 11.08598.