How to display shorter version of wp_editor?

You will want to pass some parameters into your $settings array for wp_editor() that only tells it to display the quicktags for strong,em,ul,ol, and link per your screenshot.

I didn’t have time to test it, but this code should give you want you want:

$settings = array(
   // Any previous settings you are currently using     
   'quicktags' => array( 'buttons' => 'strong,em,ul,ol,link' ),
);

wp_editor( '', 'my-short-editor', $settings );

You can replace 'my-short-editor' with anything but keep in mind this will be the id attribute for the resulting editor when rendered.