That "Start Charging" button exists on both the app and in the charging screen on the touch screen in the car. They both do the same thing. I use them both, and they both should work.
So that is the right method. I'm just not sure why it isn't working for you. I'm going to ask a question that may sound obvious. Since it immediately says "Charging stopped" is it already at the charging limit you have set? If it's at the limit, it will not charge above that, no matter how many times you click the button that says "Start charging". So maybe try moving that limit up, and then see if you can get it to start charging when you push the button.