right, in that case then....
I had a biter for a child, by eldest son (I have 5 children, all now 8 - 16yrs old, eldest son is 10).....I hated it, was the worst feeling in the world parenting a biter. I think you know this, you have already stated your elder DD bites/bit.
I NEVER appreciated anyone else telling him off in a shouting way. remove him, hand him to me to sort out while the other child is dealt with, all ok, just don;t yell at my child and storm out as it confuses hi, he might have stopped but it was shock stop, not understanding shock (I hope you are getting what I mean here).
and to react in a large family setting....sorry, I would have reacted the way you say you family did towards you.
now, I actually appreciate other people telling my children off....they take if better and more often than not, understand btter and I think nowadays they are rather too adept at tuning my voice out due to hearing it too often!
I do completely understand that you reacted in a mother lioness way, and protected your cub, but to react the way you did.....very OTT IMVHO.
that said, not one of us here is perfect, no one ever is, and parenting is damn hard, and we make mistakes all the time, and learn from them. you will learn from this, and hopefully so will your SIL and neice.
but next time, leave the shouting to them, and just walk away and sort out your own DD.