You won't be able to move the segments into the media browser, because the original clip is already there. Nor can you split a clip in the media browser. That must be done in the timeline. You can duplicate the original clip in the media browser, and trim back the duplicate, that would leave you with the original clip and a segment in the Media browser. However, as explained below, the segment will still contain the entire movie that has only been trimmed back.
When you split a clip in the timeline, or trim it in the browser, you are creating a segment that still contains the entire movie, although only the split off part is visible. You can drag out the edge of the segment to make the entire movie visible again.
To create independent segments that don't contain the entire movie, split the long clip in the timeline and, for each segment created, select it and do an Edit/Copy and then Edit/Paste it into a newly created project. Then export the new project to your desktop. You now have an independent clip that contains only the segment and not the entire movie. Import the movie back into an iMovie event. Do this for each segment and import them back into the same iMovie event. Then, when you open the event from the iMovie project sidebar, segments will collectively appear in the media browser. You will also have each of the projects that you created for the clips, to keep or delete as you wish.
It sounds a bit complicated to explain but it is actually fairly easy to do. After the first time, you will have it down.
-- Rich