Personally, I would choose a well broke horse over a pony. I don't want to offend any pony owners out there- both are great animals and both have their advantages and disadvantages. However, ponies seem to typically have more mischievous and unpredictable personalities. Once again- this is just my experience, animals in general can be unpredictable.
Also, the nice thing about a horse is that depending on your daughter and the horse in question, she will be less likely to outgrow it.
My parents bought my very first horse for me when I was twelve- she was 21 years old, pure Quarter Horse, and a 15.3 hand retired cutting horse. She took very good care of me, and taught me more than any riding lesson ever did.
If it were my child, I would feel safer with her on a horse with lots of experience and a calm demeanor.