That sounds awful, and I'm sorry to hear your daughter is being difficult.
There are already suggestions for getting professional help, but I'm going to focus on exploring some options which don't involve a professional. Of course, involving a professional is entirely up to you.
It's okay to be angry with your daughter. Anger is not abuse.
Bear with me here for a second. Children have less nuance than adults. Adults tend to look down on each other for getting angry or showing "negative" emotion. If you're angry, even very angry, show it! Explanations and reasoning with children only go so far, but your daughter will instantly understand she made you angry. But...
Whenever I try to respond to her behavior with negative consequences, she has huge meltdowns.
Those huge meltdowns are part of her misbehavior. Children and adults get emotional when they realize they have broken the rule and are caught. And they often try to talk their way out. Or grovel their way out. And, for your daughter, so far it's worked.
Which brings me to my second point.
Your daughters behavior continues because it's worked so far.
Children are fast and selfish learners, and can be practical to the point of being calculating. This is why you question:
Who is this child? Is she a nice, smart girl? or a deeply disturbed psychopath?
She is a very smart girl, who is sweet and nice at school, because it works to get what she wants there (approval, respect), and is mean at home because it achieves her goals there.
If you take away whatever reward she gains from her behavior at home, or change the means she achieves her goals, you'll change her behavior.
Just an observation, but approval and recognition sound very important to your daughter, and whether she gets it through positive or negative means doesn't sound important to her.
What can you do?
Be direct.
When your daughter tells you she hates your cooking, tell her in no uncertain terms that what she said hurts your feelings.
Change tactics.
Ask her what you're doing wrong, and listen. (Remember this is a tactic, but also be genuine.) Take it all with a grain of salt, she's six, but also consider that, if your daughter is being genuine, she is expressing her real feelings and reality, which may be very different than yours. Ask her why she says/does the things she does. Tell her you love her and you help the people you love... how can you help? If she won't talk, ask her why? Really try to figure her out based on her own words/thoughts.
Simple, firm rules.
Have rules which are always enforced, and you should never feel guilty about. Like:
Tantrum time limits. After so long, say 10 minutes, she's restricted in her room. Count down. Restriction lasts until she stops crying, never before.
Dinner with thanks or no dinner. See rule one of this a problem.
No desert or treats for bad behavior, but cookies for good behavior. It's very Pavlovian, but powerful.
And maybe a strange suggestion, but it's worked for me.
Buy large headphones and a child gate.
If your daughter simply can't be contained, ignore her. I play video games with headphones on, door open, and baby gate closed, until the tantrum is over. This way I can keep my eye on things while they are still distant echoes, literally. Watch TV with the well behaving children and husband, exclude the misbehaving child. This works much like a cookie: she gets your attention only when she behaves well, and not when she's poorly treating you.
I feel strongly I need to caveat that method with, once the child becomes coherent or makes reasonable demands/requests, I immediately listen with my full attention.
Anyway, you can handle this. Don't get too frustrated with yourself or your daughter - you'll get through all this.