What setting enables split screen mode in OS X El Capitan?
Split screen will only work when you turn on 'Displays have Separate Spaces' setting.  

After you turn this on, you can use both methods you described.

To expand on bullet 1, from your desktop (i.e., not in a full screen app), click and hold on the green circle and it will give the option to put a window on the right or left.  It will then prompt to put a second, already open, window on the other side.  The two apps will now be in full screen mode together.