I've already given one answer, but I wanted to give another based on one thing you say which you suggest may be at the heart of what's confusing you. First, let me set up an example which may help clarify:
Consider someone working out at a gym. Their workout is based on what in gym terminology we might call a "circuit". It might look like this:
- one set of five bench presses
- rest for 30 seconds
- one set of five squats
- rest for 30 seconds
- one set of five deadlifts
And then their gym session as a whole might consist of three of those circuits, with 90 seconds rest between each.
Now referring back to the Merriam Webster definitions, it might be tempting to say that "cycle" is simply a synonym for "circuit" in the above. But in fact the MW definition is such that "cycle" refers very specifically and only to the time itself, and not to the various exercises.
So the thing you say that may be relevant is:
It's weird though because things that are happening during the duration of the cycle are also the cycle, but the definition seems to me like it only talks about the time between.
But that's not exactly correct. It is not the case that "things that are happening during the duration of the cycle [i.e. the five items in the circuit] are also the cycle" Applied to the gym example, the word "cycle" means nothing more than the blob/lump/space of time from the start of the first bench press in a circuit, to the last deadlift in that same circuit.
Also, it's not clear, in the above quote from your question, what you mean by "the time between" Do you mean the time between exercise within a circuit -- i.e. 30 seconds? Or do you mean the time between circuits -- i.e. 90 seconds? (Or maybe even the time between each benchpress, each squat, etc -- i.e. not specified, but usually very little if any time!)
In summary, using my example:
Interval could correctly be used to refer to any of the times I mention: the 30 seconds between exercises, or the 90 seconds between circuits. The key is that both have meaningful start and end points
Cycle could correctly be used to refer to any one sequence of the five items in my list. The key is that the entire sequence of five is subject to repetition.