Sorry, OP (if you're still there!) but I have had a couple of "mummy-friends" who I have diplomatically pulled away from due to their children's behaviour. One of them in particular reminds me a little of your description. Her ds was much wanted and a result of long years of fertility treatment. She did try, and she was a really nice person, but she just never seemed to be "in charge" of the situation, for want of a better expression. She often wheedled him, and asked other children to let him have what he wanted, just for the sake of peace. She got very flustered by his behaviour, and made lots and lots of excuses for it, but seemed to not really address it (not saying you are like this, btw)
I liked this person, but not enough to endure playdates with her ds. You do have to look out for your own child first, and mine didn't want to be hit/ slaped/ kicked/ have their toys taken away. It was awkward when she (the mum) got all upset and flustered, noone quite knew what to say.
I would say all my dc are "spirited"- who would want to describe their child as lacklustre?? I'm not convinced putting labels, of any sort, on children is productive. Your link could describe all 3 of mine, although they are all very very different!
Some mummy-friendships are, in honesty, friendships of convenience, in the nicest possible way. I keep in touch with just a few of the mums I met when the dc were little, the ones who I relate to well as individuals, and we usually now prefer to meet up without the dc. Don't worry about this friend- let it go for now. You will make many more friendships as your dd gets older, and not all will stick. Your old friends will be there too, when you emerge from the "mummy years"! Hang on in there 