Puppy is scared by barking

My 4 month old flips out if we're on a walk and he hears a dog bark. He eventually became okay with our immediate neighbors' dogs barking when he's outside at the same time as them (he runs back and forth behind the privacy fence where he can hear them; he seems to think they're being playful, but they're older terriers and definitely don't sound happy). But all other dogs cause him to try to run inside as quickly as possible and can end our walk before it even begins. Earlier, we stepped out of the house and a house two lots down had their dog out back and it barked, causing my puppy to immediately run and scratch at our door to be let inside.

He also doesn't seem to like when we play videos with dogs barking. But again, he's fine with our neighbors...

Most of the examples I can find are people complaining that their puppy barks when other dogs are barking, but my puppy has maybe barked 5 times ever and never in response to other dogs/barking (pretty much has always been due to my partner getting him riled up during play, during which he gives one singular bark). He definitely has a "fleeing" fear response. I'd love to be able to walk him without having to end our walk the moment a dog nearby barks, but I can't get him to calm down once he's heard it.