What is the difference between setDisplayHomeAsUpEnabled and setHomeButtonEnabled?

For what you want to do, actionBar.setDisplayHomeAsUpEnabled(true) is enough.

For the difference :
actionBar.setHomeButtonEnabled(true) will just make the icon clickable, with the color at the background of the icon as a feedback of the click.
actionBar.setDisplayHomeAsUpEnabled(true) will make the icon clickable and add the < at the left of the icon.