I agree with the others who are a little concerned about the level of play. The video looked to me like three dogs who are quite lacking in social skills with other dogs, and your following posts do confirm that. I don't think it's beyond help, though, I believe that those puppies could learn appropriate behavior with other dogs and would be able to play with other dogs. But there are just a few problems that crop up right now:
1. The dogs' age - they are basically "teenagers" with a lot of hormones kicking in. They are also beginning to see themselves as "grown-ups", but still lack a LOT of mental maturity. At this age, even the most social dogs can be difficult around other dogs.
2. The dogs' genetic history - you are right to be cautious with them around other dogs, based on the mother's history with other dogs. You may also keep in mind that the mother may not be a very good role model for the puppies to be learning about dog interactions as she is not good at it herself. I'd suggest limiting the amount and type of play the puppies are allowed to engage in with the mother; not to say they can't be around her at all, but keep the play down to a minimum.
I think you are right to limit the off-leash play the dogs get with outside dogs at this point, and if you make the choice to just avoid the problem altogether, that may also be a good solution. But if your dogs are going to be around other dogs, even on leash, they do need to learn better social interactions with other dogs, and one of the best ways to teach this is with off-leash play.
I would suggest reading the book "Click to Calm" by Emma Parsons; the tips in the book will help you with on-leash interactions and will help you learn how to train your dogs to be appropriate around other dogs, which is a great starting point. The sooner you begin this training, the better.
If you do want to work on off-leash play, I'd suggest finding a trainer with a good neutral dog that can help you learn what is appropriate with your dogs. Because of the dogs' age, I might recommend just working on leash (with "Click to Calm") for the next year or so, and when the dogs are fully mature (18-24 months old) the off-leash play might become a lot easier. Hard to say for sure, though, so I would definitely defer to the trainer you choose to give her own advice.