D.B.
A lot of people think that potty training is about LEARNING - and that, therefore, a "very bright little girl" should learn it quickly. Unfortunately, that's not the case. She has learned what you expect, so she's excited about the big girl underwear - but she knows she can't do it, so she runs away. She sits on the potty because you want her to, but she doesn't "go" because she doesn't have to, hasn't figured out how to push anything out, and so on. She's sitting there, but her brain isn't sensing anything, so nothing happens.
Here's the problem - it's a developmental stage. Her brain has to get the "full bladder" or "full rectum" signal from those organs, and she has to translate that into action. That part of her brain isn't developed yet. It doesn't mean there is anything wrong with her; it just means that kids have different timetables. Don't some of us have early talkers, while others have early walkers? Don't some kids read early, while others develop incredible gross motor skills sooner than their peers? We cannot rush them - we can provide them with opportunities and encouragement, but we can't sit there with a 9 month old and say "You're bright, why don't you walk???" just because a friend's child did just that.
Your daughter isn't ready. There's nothing you can do except celebrate and be confident in her other achievements and milestones, and wait for this potty stage to develop. Put the diapers back on, forget about the "training" part, and take the stress away from both of you. Tell her, "when you're ready, we'll try again." See what happens in a few months.