How can I fix my technique and catch the birdie higher?

With most shots, you don’t position yourself straight under the shuttle. If you look back, you will notice that you are slightly to the left of the shuttle’s path. You can reach higher if you are straight under, and in fact it will also help you to hit with better technique.

For most players, forehand shots played from closer to the backhand/overhead side feel more natural because it automatically forces you to stretch more, resulting in a higher hitting point.

Try to position yourself more straight under the shuttle, so a little more to the right. You want to be looking straight up to the shuttle when it flies towards you.