Wow, there are some unhinged posts on this thread.
OP, I can see wrong on both sides here. For context, I am a dog owner, and our dog is reactive after being attacked twice by off-lead dogs who were not under control.
I would not recommend having your dog off lead while you are also watching your children. An off-lead dog should be monitored all the time, as you have no physical control over it, only verbal, so you need to know if a situation is arising that you need to respond to. You will only know this if you are paying attention.
I would also not recommend having your dog off-lead unless you have faultless, 100% verbal control of your dog. This is the very definition of having your dog under control. If you do not have this, your dog is out of control. Your post implies your dog has good recall, so I would imagine if you had been watching, you would have seen your dog start to approach. At this point, stopping your dog and checking in with the man before allowing your dog to engage would have been the right thing. Yes, his dog was also off lead, but his dog did not approach your dog - it was the other way around. So he may have had his dog under close control, while you didn’t. It was your dog approaching that started the sequence of events that ended so badly.
On his side, he should accept half the responsibility. If he didn’t want your dog to play with his dog, he should have (if he had full verbal control) called his dog away and asked you calmly to recall your dog. He didn’t, so I assume he was happy for your dogs to play together until the point his child was hurt. That means he was equally responsible, and I’m sorry that you were so frightened and intimidated by his reaction. It was out of proportion, considering he was also to blame. Playing dogs are daft, they don’t pay attention to what’s going on around them. That’s why I wouldn’t let dogs play near small children - it’s an accident waiting to happen.
As dog owners, it’s so important that we have full control of our dogs at all times, and we have to take responsibility for our dogs’ actions. It’s always better to be over cautious than under cautious. I hope you are okay, and can take something from this to help you avoid this kind of thing in the future.