You should always keep in mind a thought that I find very helpful - Aggression is never the problem, Aggression is only an outcome of a problem.
Based on what you described, I recognized fear, low self esteem, nervousness and bad social skills. The fact that the dog is able to play and socialize after a certain point is a very good sign. But you have to understand that dog parks can be a very bad place for an unstable dog to socialize. Always bring your dog to a dog park after good physical exercise, so he drains some of his bad energy (fear, anxiety, etc.). This is what every dog owner should do before bringing a dog to a dog park imho. You have to master the ability to read your dog and his surroundings and you have to enforce a protocol how other dogs approach and meet your dog. There is nothing wrong that you ask from another dog to approach your dog very calmly and respectfully - this is the key aspect that your dog is looking for. You are your dog's teacher, mentor and protector and with him being unfortunately unstable at the moment, you have to provide good experiences. Try to connect with people that have stable obedient dogs and have him socialize with them, they learn from each other very quickly.
Be careful before letting you dog in a dog park, read the other dogs first. Develop your instincts, because other dogs will always gang up against an unstable weak-energy dog (fear, nervousness, anxiety based aggression are all weak energies in the animal world). Your dog has (for now) bad social skills, he will regain confidence after a lot of good experiences. You have to provide those experiences, meaning you have to give him a good chance - as I said, dog parks are (usually) an unstable chaotic situation. The ultimate challenge for a dog is to remain stable in an unstable pack of dogs. So that is asking too much for now from your dog.
Good luck, hope this helps some
...ps. I'm talking about dog parks, because you mentioned organised dog meeting which can be very similar.