As usual, I will state my lack of professional experience here, but I've been doing bridges since I was a child, so I can speak from some personal experience.
If I understand correctly, your goal is increased backwards flexibility to allow for a more severe arch. First off, flexibility in general is, to some degree, inherent. You may need to accept that there may be an upper limit of what you can reach there. Secondly, while it is a whole-body movement, there are three major points of flexibility to work on, and each person progresses differently on them. The two pictures you provided actually provide a good reference.
First of all, we have both the most obvious area of flexibility and, unfortunately, one of the most limited, the back. You'll notice that, in the images, the back has a fairly similar arch. That's because there are some fairly inherent aspect of human back flexibility. There are some people who have more flexibility, but most of us can only get so far before we're limited by the structure of the backbone. The primary area where you will be able to improve there is actually on the other side, the abdomen. Holding a proper bridge does require core strength, and at first, one of your limiters, especially for the Standing Chakrasana, will be the dynamic flexibility of the abdomen and the ability to slow your movement using that core strength.
Secondly, there's the hip flexibility. This can primarily be seen in how close the hips get to a "standing" position. In the second image, you can see where the legs look almost like the person is standing. This is a matter of pushing the hips forward toward the ankles.
Thirdly, and the one where many people are most limited, is the shoulder flexibility. Notice how, in the first image, the shoulders are driven out past the rib cage. In comparison, the second image has has the hands, shoulders, and ribcage all in one line.
And here's where I apologize because I've been answering everything but your question, because I felt you needed the background. For building flexibility, you will primarily be doing the "sleeping" Chakrasana, with your focus being on working your hands closer to your feet while keeping your core firm. As I stated before, the two main areas of flexibility to work on are the shoulder and the hips. To specifically work on those, you will basically alternate shoulder and hip flexions, first pushing your hips toward your feet, like you were about to reverse the "standing" motion by sitting up, and pushing your ribcage over your shoulders and your shoulders over your hands like you were about to kick up and over. From personal experience, this is a good one to have a friend help you with, keeping one hand under the small of your back to provide support. The movement applies more strain than you might think, and you especially do not want to lose your arm strength and drop yourself on your head.
So, that leads to the question of, where will the "Standing" Chakrasana help you out with your flexibility? First of all, because you need to control your descent, it's good for building your dynamic core strength and flexibility. Secondly, because it has your body weight carrying you, it can sometimes be useful to break through a small plateau in your flexibility. I personally recommend the "Standing" version for general exercise because it is more of a whole-body movement that requires a comprehensive opposed movement and because it's easier to "start" with the hands and feet close together as compared to the "Sleeping" one where it's entirely pushing up into the arch, and it's difficult to start with the hands and feet close together since you start from flying on your back.