You now want to turn right at these traffic lights with a green arrow sign. What is the right course of action?

You answered correctly. At a traffic light with a green arrow on a sign, you must first come to a complete stop at the stop line, even if the way looks clear. After stopping, you need to make sure that other road users—such as pedestrians and vehicles—are not put in danger or obstructed before turning. The green arrow here does not allow you to turn without stopping or just slow down; stopping first is always required by the German traffic rules (StVO §37). This ensures safety for all road users.