This isn't a contradiction at all. GP is saying it isn't proven to protect others. Your CDC quote says it "may" protect others. I don't know the actual science

This isn't a contradiction at all. GP is saying it isn't proven to protect others. Your CDC quote says it "may" protect others. I don't know the actual science here - but they could very well have zero evidence that it protects others and still correctly say that getting the flu vaccine may protect others.