S.A.
EDIT...
I just wanted to add that I'm a mother of an only child but grew up with multiple siblings (5 brothers and sisters and a niece the same age as my youngest brother) and this stuff DOES happen, and fairly often, especially with 2-3 year olds!
ORIGINAL:
I think with your daughter you handled it appropriately - both you and your husband communicated that it's not okay to bite and why (because it hurts her friends when she bites them). Punishing her showed her that there are consequences to her actions when she does things that aren't okay. You might also want to teach her other ways of dealing with her frustrations since she does have a history of biting.
I used to work in a daycare as a teacher for the Toddler room and as assistant teacher in the 2-3yr old room, and biting is VERY common behavior - for the very reason you pointed out - frustration! It's such a tough time at this age sometimes.
I'm a fairly "old-school" parent, despite being one of the (if not THE) youngest mothers in my son's class (I'm 28, he's 8 and in 3rd grade). When my son bit me for the first time out of frustration, I bit him back right away - and hard enough to leave a mark. He had bit before while playing and I had told him "No biting, that hurts mommy!" but this time was different. After that, he NEVER bit again (and he was younger than 18 months when this happened). Maybe your daughter doesn't realize how much biting hurts? Has she ever been bit by another child? If that happens (or happened), make sure to make the connection between her pain when she gets bit and the kids that she bites. At three, she might not grasp that on her own yet.
As far as the friend, she doesn't sound like a very good friend. From what you've said, it seems like you handled it well - you accepted responsibility for your daughters actions (some parents try to deny that their child would ever do such a thing!) and you apologized. This other mom seems to be over the top. Personally, if I was in your situation, I would discontinue this friendship because it's not healthy for you to feel like you've got to defend your parenting to this mom. Her threatening to call the school and make sure they know about your child's biting habit is too much. You should make them aware of it yourself if she is already enrolled but also make sure they know it's a behavior that's not acceptable in your house and it's something you are working on with your daughter.
If the other mom apologizes for over-reacting and you want to continue the friendship - go for it. Personally I evaluate friendships on what I get out of them - if I don't get some sort of happiness out of having a friendship, then what's the point? Evaluate this relationship with this woman and determine if it's one you really want to keep having, then act accordingly.
I wish you the best of luck!