I have a butterfly with cycle layers flapping the wings. This is going continuously. I then created another sibling group with the butterfly landed (wings not moving). I copied the wings into that other group. There is then a swapset trigger to pick between the continuous cycle layers or the static unmoving wings. You can see the cycle layers behavior on the Flying / Right Wing and Flying / Left Wing layers, but none on the Landed / Right/Left Wing layers.
Reading the docs, "Cycle Pause" seems to be a tag you can put on one of the layers to stop cycling at that layer. So you can make the sequence trigger from that point when you trigger the cycle next time (or when you release the trigger). I have never tried that however.
There is a blog post and video on the butterfly here: https://extra-ordinary.tv/2018/02/04/butterflies-and-poppies/
First, I did not mention the "Start" = "Immediately" property to make the cycle layers start without doing anything. (It was in the screen shot.)
But I just had a quick play with the cycle pause tag. Well, I made it do something, but I have not worked out why its useful! E.g. I created a cycle layer, put the "Cycle Pause" tag on a layer, then added a keyboard trigger. What would happen is the keyboard trigger would make the layer appear (hidden otherwise!) and it would stop on the layer with the pause tag. Clicking the key again would make the layer disappear. Clicking again it would appear and play the sequence again. But having the layer disappear did not seem useful. So I can confirm (1) it does do something and (2) I have no idea how to use it in a useful way! :-)