Hi J.,
I guess my response is totally opposite from the others. I don't think 2 years old is too young to potty train. I speak from experience. I trained my daughter starting at 18 months. By the time she turned 2, she was completely potty trained.
I know that each child is different. However, in my opinion if your daughter is showing signs such as you described she is ready. Especially if she can show you where to go to potty.
If she is refusing to wear diapers, then she is understanding a lot concerning the potting issue. Tell her that she has to either use the potty or put on the diapers. To be honest, if she is staying dry all through the night, that alone is a huge step. Since nights seem to be the worse when potty training.
Try taking her into the bathroom when you go. That was something that helped my daughter. Kids like to do what they see their parents do, so this wasn't anything different. I would also take my daughter to the bathroom about 20 minutes after she ate. That helped a lot as well.
I am not big on the rewards system. Sorry. I gave my daughter lots of praise when she used the potty and that was good enough for her.
I had to take control and not allow her to control me. I didn't force her to use the potty or scold her if she had an accident. Patience is the key. I hope this helps. Be blessed!!