That is indeed because the child is traveling as a lap child. It's an age-based fare, so you have to prove it.
The OP has a child that is over 2; no more lap baby status for you when you turn 2. So unless there's an airline with a child's fare that is actually good and the OP has it, the domestic flight will NOT need proof of age.
Disney parks are pretty good about figuring out when a child is under 3 and over. Even a tall child still acts like a little kid. Even an articulate child still has something "babyish" about them.
I can think of just one child that I know out of all the children I know, who might have caused problems at the Disney gates, because she is tall and big and GROWN looking, and is INCREDIBLY articulate and bright and smart and I don't think she looked like a baby even when she WAS a baby. I met her as a 3 year old and she seemed to be 6. She's now 6 and I haven't seen her in person in awhile, but pictures on her dad's FB page just floor me. But that's one kid out of all the ones I know (luckily her father wouldn't do anything as humdrum in his opinion as to go to Disney). Disney is pretty good about it all. They also know that kids that age like to "age up". My son at 3 was saying that he was something like 45.