B.,
I am going through this with baby #4 right now who is 18 months old. With my first three, I waited until after their second birthday to potty train. Everyone said it was too early, but my maternal instinct said I had waited too late. They all potty trained within a day to a few weeks with this method.
With number 4, I started putting her on the potty after dinner while I ran the bath when she was 12 months old. She started peeing in the potty about 50% of the time. Now she is 18 months and will hold it after dinner until I can get her to a potty and pees in it daily. She tries to pull her pants down and even finds the potty and sits on it when she has to go. I started when she was so young that I didn't have any resistance from her.
I don't think I can fully potty train her at this point because she is still so little, but I intend to do it as soon as the weather warms up and right now I just let her pee on the potty a few times a day.
A few generations ago, it was normal to potty train between age 1-2. In some cultures, it is even earlier. (The Amish have their kids out of diapers by 12 months and the moms just take them to the potty until they can do it on their own.) With the introduction of the disposible diaper, moms lost their motivation to potty train. It is just easier to keep them in diapers longer. Then the kids get so used to it, it becomes hard to train them and they are often age 3-5 before it happens.
If she is interested in the potty, let her use it. Pediatricians now say "don't potty train early" not because you shouldn't, but because they don't want to deal with all the phone calls from frustrated moms who get into power struggles with their toddlers.
I used the book "Toilet Training in Less than a Day" and it worked great with my first three, although two of them took longer than a day. With child #3, I had some practice with the method so it only took 4 hours to fully potty train at age 26 months. (He was my oldest to potty train.) You can get the book on Amazon.com or on eBay.
Good luck,
S.