The rise in CO2 isn't the cause of the warming, it's the effect. But the difference is a matter of a few centuries, a fact easy enough to hide when your diagram purports to document change over half a billion years.
You are correct. The CO2 levels rise due to warming. It has to do with the solubility of CO2 in water. Warming changes this balance. The warmer the ocean is the less CO2 it can keep in solution. Just like a warm bottle of soda. Open it and reduced the pressure and the CO2 bubbles out.
Oddly the Vostok cores from the Russian station in antarctic proves this. The Vostok cores were used to prove global warming due to CO2 levels. That was false. The Russians initially used rather large intervals for analysis of ancient CO2 levels. It proved conclusively that CO2 was the driving force of global warming, case closed but it was not. The Russians then analyzied the cores on a smaller basis of interval of time. Voila! They discovered that temperature rises first followed by CO2 rise. It is all about the solubility of CO2 in water relative to temperature.
The warming and cooling of the earth are a function of celestial mechanics of the earth's orbit. If one looks at the past geologic history of warming periods greater than our present great beasts roamed the earth. They were big because they had a lot of food due to CO2 levels multiple times greater than today. CO2 is plant food and vegetation flourished in the extreme. It feed the food chain all the way up to Tyrannosaurs Rex.