This lawsuit is over a Freedom of Information Act (FOIA) request. FOIA is a law requiring government entities to release certain types of information upon request.
Plaintiffs asked the CDC to provide evidence of a certain claim. In response, the CDC provided 20 such documents. Link 3 in the OP is to the agreement by attorneys from both plaintiff and defendant dismissing the case. The articles after this are simply arguing that the scientific evidence provided is flawed - this isn't anything a court decided, just their opinions which do not seem to be based on science - their claims about the CDC "losing" a case are not founded.
In summary, the case was indeed as in the title of the question here: to "produce evidence that vaccines do not cause Autism". The CDC provided the evidence, which was the goal of the lawsuit. At that point, there is no more case. The CDC could have argued that they weren't legally required to provide the evidence, and then the court would have to rule whether this was correct or not and perhaps order them to provide evidence.
This would be like if you went to court asking for someone to give you $1 that you are owed. They agree you are owed $1, and offer to give you $1. The case at that point would be dismissed with an agreement signed between the parties stating that the $1 has been given. Afterwards, someone writes a blog post saying that US currency is not legal tender. Such an argument does not change the outcome of the case, it's an orthogonal argument.