Hi - this is an interesting issue.  You may have to provide more information because it seems the issue is either errant behavior in your install of MEP, or you are not understanding something. 

What version of MEP are you using?  If all else fails, try the old uninstall/reinstall routine, especially if you are running a version prior to MEP 14+.  I've never had to do that with version 14, but the prior versions occasionally did weird things that I could fix only by uninstall/reinstalling.

Assuming it's not MEP, I'll make a brief description of splitting scenes, in case it might help you.

The "cursor" is the red time line.  Place the time line at the point in your movie project where you want make the split.  Find the "Split Scene" function and click it.  In version 14, the Split Scene button is the first button to the right of the "fast forward" button (under the video monitor window), also, it's under "Edit" "Split Object" or you can just hit the "T" letter on the keyboard.  It's likely the same in your version.

The red time line should stay right at the point of the scene split and not move anywhere.  If it does, it sounds like errant behavior.

Hope that helps.

Tim