I don't know if anyone has suggested Dr. Deborah Ruf's book to you yet but you should definitely get a copy of it. She has recently changed the title but it use to be 'Losing Our Minds'. She goes through the levels of gifted and signs to look for at different ages. It really helped me get past some of the denials.
This said ... I have been here with my DD (a recently turned 3 yr old). She is a very verbal child and learned a lot of the basics way before she should have and yes the books were around but we certainly didn't make it an agenda for her. As for the isn't it possible that it is just because the other kids have not been exposed to it ... I have a few comments and I hope it doesn't come off as negative. I have a friend that has a daughter who is 2 months younger than mine. We talked all the time about what the babies were doing which led to me informing her of what DD was doing ... she in turn would add it to the curriculum for her daughter. It is night and day the differences between the two girls. And for another example... my DD has been in dance since turning 2 and the moms talk. On a few occasions the conversation turned to how they were frustrated when trying to teach their 3 yr old daughters colors. For most normal kids it takes numerous repetition for them to get the concept. I had a woman on another board give an example that resonates with me about the differences. She talked about teaching a bedtime prayer. The normal child will be able to add one or two words each night but still stumble on it while the gifted child will memorize phrases of the prayer and have it by a couple of nights. I smiled the other day, thinking about that when my DD clearly had learned some prayers at school. It was her second day at preschool and she was reciting the whole prayer.
And the other big sign of not 'normal' at this age is being able to reuse the information later on, such as the planet example. One time we were driving home (DD was around 2) and she asked why the trees looked dead. I explained to her that they were dormant and would be blooming again in the spring. That night she informed me that she needed to become dormant to blossom in the morning.
I hope that helps and definitely get a copy of the book. It really will help.