I do press the binding in half before stitching, and I press the binding away from the quilt all the way around before turning it to the back or front (depending on which I am doing). I saw a video from Donna Jordan where she did that and it made a great difference in how the binding folded over, she uses her finger, but the iron works better for me. I also fold the corners and give a press during this step.
Here is a link to her video and if you want the entire binding segment starts about 23.51 and the finger pressing technique at 29.07. This entire video is filled with great ideas and hints.
https://www.youtube.com/watch?v=SE5CqpWaReA&t=1178s