Under either set of rules, the order of events makes no difference. If a child is a permanent resident and under 18, and then the parent or parents naturalize, the child gets automatic citizenship. If the parent or parents naturalize and then the child gets permanent residence, the child becomes a U.S. citizen the moment he or she becomes a permanent resident, if that happens before the child is 18.
If you are believe you are a derivative U.S. citizen, the best proof is a U.S. passport. It costs you less to get a U.S. passport than a Certificate of Citizenship. In addition, in my experience, passport officials are more fair that the USCIS in deciding whether a person derived U.S. citizenship. A U.S. passport is valid for all purposes in proving U.S. citizenship. In any event, you are going to need the passport to travel.
