Walking away should have been the first thing she did to be honest.
How is walking away going to help? Surely the greyhound would just regard that as part of the game?
It has the opposite effect, it's basic dog psychology, if you don't remove yourself, an excited dog that has approached you will focus all it's energy on you/your dog, it was pawing at her dog and hit it on the head, hurting it, because she was standing there allowing it access, if she had given a firm NO, then turned and walked her dogs away it would not have been able to do that, it is also giving a clear signal to the dog that you are not interested in playing with it.. dogs aren't mindless beasts, they understand commands and dog and human body language, it may have followed, and may have tried to interact, but in the process of walking you can block it from getting near your dogs, if it is not getting any excitement in return most the time it will just give up and move on to the next poor person minding their own business, I've had to walk TOWARDS an owner before to help them get it back, but the movement and blocking stopped it getting access to my dogs.
Standing there shouting, kicking out, dancing around with your dogs in your arms, turning round in circles (I've seen people do this so many times to know exactly how OP reacted) it is just a massive game to an excited dog.. it means absolutely nothing to their dog brain apart from fun, and will cause them to jump, pounce, act nuts basically unless there is an owner nearby to grab it, which there wasn't, it can cause the picked up dogs to act more aggressively too from a position of elevation/safety. That dog would never have jumped up OP if she hadn't have lifted her dogs in the air, it escalated the situation enormously, which is why she had to use violence to end it. Lifting the dogs should be a last resort if you think it will save them from being bitten/attacked, which was not the case here at all.
I've had dogs all my life, and one of my dogs now has been jumped on so many times by idiot dogs of idiot owners that he has zero tolerance for them anymore, it's a shame as he used to love all dogs, but now he will snarl, snap and see off any off lead dog that pester him, most people follow the etiquette rules around dogs on leads, but we still come across idiot's like OP did, and 9/10 times turning your back and walking away causes it to quickly lose interest, sometime I have to shout at it too (normally performative so the owners get the message), but only once in 30 years have I had to be violent to a dog and that was actively attacking.
OPs dogs are reactive too, and her nervousness about approaching dogs will not be helping that, so she needs to find a much better, safer, strategy for dealing with unwanted attention, and learn how to calmly de-escalate, rather than follow her instincts which actually escalated things, it shouldn't happen at all but it just does unfortunately and as a dog owner you need to be prepared to deal with it effectively.