Hmm, so from my experience with my own dogs, I can tell you: big individual differences are totally normal. With two-year-olds it's similar - some still need a nap, others don't. But honestly, I'd watch it more carefully. If your daughter gets up at 6 and just isn't tired at noon, that can have several reasons. Maybe she's just a kid who needs less sleep earlier on - some kids are wired differently. But I'd also question it: Did that change all of a sudden or has it been like that for a while? And how's her behavior in the afternoon and evening - is she overtired, irritable, or does she actually seem fine?
What I'm skeptical about is that sometimes people just give up too early. Some people think because the kid doesn't fall asleep right away, it doesn't need sleep anymore - but that's not necessarily true. Have you really tried it consistently at the same time, with set routines? With my dogs it works best with structure and patience. That said: If you really try it for several weeks straight and nothing works, it could just be that your kid is in a phase right now where it doesn't need a nap anymore. That's most normal from about 2.5 to 3 years old, but not impossible earlier.
I wouldn't see it as a failure if it doesn't work out. But maybe try it for two, three weeks with real structure before you give up. And the daycare situation is different anyway - other kids are involved there, it's more tiring, so some suddenly sleep anyway.