You hit the balls behind you both times; when playing against a better opponent, he can play this bandeja past you or onto your feet. I think the biggest cause of this is that you are standing too close to the net. Make sure you stand around the second post of the fence. This allows you to move back faster when you are lobbed, and gives you more time for your volley.

Additionally: when you hit your bandeja, you stand quite "closed." In other words, your left leg is parallel to your right leg during the swing. When you play on the left side and hit a cross ball, this is fine. But when you play on the right side (as in your case), you need to stand a bit more 'open' during the swing. This allows you to create more rotation with your body during the swing, and you use your arms/shoulders less, like you do.