That link was bit of fun , chill out man ( or in your case, warm up ..)
I didn't say "nothing to do with the Sun". It should be obvious that the study actually looked at the effect of cosmic rays on cloud formation/climate change, the Sun can have an effect on the amount of cosmic rays reaching the earth. So this study may rule out one indirect mechanism.It doesn't rule out the Sun per se. So to say "No Sun link" to climate change is wrong and misleading, but would probably be taken at face value by Joe Public.
No, I wouldn't expect an instant temperature response to CO2 .