I think that HK, polar ear, and each have made some really good points.
TBH, an aspect of this anecdote that concerns me is this:
Later a dog ran into the group, and DD climbed a tree to escape it. She HATES dogs. She's convinced they're all going to bark and bite. She was terrified.
Not liking dogs is one thing, but I'm concerned about believing that ALL dogs will bite, and fear of barking. Sure, there are dogs that may bite, but most don't, and I think it's important to be able to distinguish aggressive behavior from friendly behavior. The world is full of dogs, and I think it would be very difficult to go through life being afraid of animals that are everywhere.
I also understand that having a negative experience with a dog will affect a person's outlook on dogs, but this doesn't negate the need for the skill I mentioned above.
