You need to learn some coping strategies.
Such as at the time of event:
"Darling don't interrupt when I'm speaking."
"Darling we'll discuss this later. You're being rude."
"No, I'll explain it to you later."
"Come outside now."
I think if you're anxious you need to put a time gap space before you speak to your daughter, to collect your thoughts.
You've managed perfectly well on this occasion to think through why your daughter got it wrong - which means you can explain it to her later acting as the adult and parent.
You can explain what's rude and disrespectful.
At age 8 you can also explain diplomacy towards all people, as well as family.
You should also know that generally all mums support and reassure each other when dealing with their children in a group.
HOWEVER, what concerns me most is a lack of respect towards you in your family.
Your partner called you a "weird little bitch"??????????
Children pick up on and watch every little thing, including body language and exchanges between adults.
They were inside the car - you know with windows - and even if they didn;t hear (they did) they saw your husband treat you with complete disrespect. Bet that's not the first time.
Where do you think they're learning their behaviour to disrepsect you?
You need some support and validation and confidence as a parent.
I've just googled parenting classes and support.
theaimgroup.co.uk offers free qualification courses in better parenting.
Family rights group also offers free advice and guidance.
I think if you read up a bit, you'd understand it's not you - that you all contribute to your family dynamic.
And if you approach it in a more pragmatic way, as in relating to text book bad behaviour and how it should be approached professionally, it might give you the confidence to be more assertive.
Willing to bet every single course or advice topic you look up - a fundamental is two parents acting as a team and supporting each other, when it comes to expected good behaviour from children.
Have faith in yourself. Your husband sounds less than wonderful as a co-parent.