I'm not sure exactly what you're trying to accomplish... it sounds like you're trying to have one button that deletes the image, and a second button that does something else?
I will hereby admit to being a complete Laravel noob, still going through the fundamentals videos, so please forgive my poor response from a Laravel point of view. But, this is an HTML-related question, really, so I hope I can help.
From an HTML point of view, you want something like this:
<input type="submit" name="update_button" value="Update" />
<input type="submit" name="delete_button" value="Delete" />
To Laravel-ize this, as I understand, you'd add something like this in your view:
{{ Form::submit('Update', array('class' => 'btn', 'name'=>'update_button')) }}
{{ Form::submit('Delete', array('class' => 'btn', 'name'=>'delete_button')) }}
There may well be a better Laravel way to do this, but I haven't gotten to that point in the tutorial yet. :)