I can SO relate! You're where I was with my guy last year. Similar scenario, I asked his pediatrician at his 18 month checkup, and she said it would put pressure on him, to set out a potty. Soooooo, ignoring my previous experience training my older two, I listened to her. And from 19-22 months when he showed several readiness signs I didn't want to "pressure" him. Shortly after he turned 22 months I set out the froggy potty he chose and he wanted nothing to do with it :( We worked at it a couple of weeks, he peed a handful of times but never told me he needed to go and I backed off thinking he wasn't ready.
We restarted a few weeks later and did slightly better, but he only pooped in the potty once. Every other time he'd hold it and sneak away to do it or wait until naptime. Would never tell me he had to poop, rarely told me he had to pee, he'd just go. I set him at certain times each day on the potty to get a rythym of going, telling him, never asking him if he had to go, because the answer was always "no" if I asked. He still didn't seem to go with the program so we stopped again and restarted 3 or 4 weeks later. The last time he got the peeing down in a couple of days, and I realized he was holding his poop. I didn't want him developing encopresis so went searching online for help and found this site:
http://www.rogerknapp.com/medical/pottytrainingrefusals.htm
The site was extremely helpful in realizing his refusal was all about control. Kids basically only have control over eating, sleeping and pooping...so he was trying to be in control. He being smart like your daughter I realized I had to be smarter. He'd never responded to stickers and charts, M&M's, Dum Dum pops or $1 store toys, for him the reward was being trained. My job was to help him get there, and I used the "power incentive," something that I retained ownership of, and he could earn use of it for 30-60 minutes by pooping on the potty. It's whatever will motivate YOUR child to go, so you need to figure out what it is and have it bought and ready for her to use as soon as she complies. With my guy he was literally running to the potty to poop in just a little over 24 hours and was fully trained (nighttime, too, totally his doing) @ 26 months.
And BTW, he never got into telling me he has to go, other than when we are out and I need to find him a bathroom, he just goes and does his business. He doesn't announce it at preschool either, he just politely taps a teacher on the shoulder so she can escort him. So maybe your daughter will be the same way, a do-it-yourselfer, too.
Read the info on the site, there's a lot of good help for little ones like ours :)