For us, it is different as we chose a different route.
I flagged his behavior via his preschool as they have a 3rd party that can come in with a behavior specialist to observe a child there and make recommendations. Our preschool had already noted some things but then the manager set up her own observation, and then got the ball rolling as she was convinced it was worth doing. They sent someone, and then the preschool organization sent another as well. All the observations came back with the same results - kiddo should be sent to early intervention and screened. He was placed directly at the Early Intervention Center which is staffed by specialists and medical personnel. They take 6 weeks for observation so they run it like a preschool and get real observations there directly and they work as a team. They gathered a lot of info from us as well. One specialist already told us that she is convinced he is on the spectrum and I don't doubt it. We're meeting the pediatrician next week and the week after we are all setting up his treatment plan.
There were other routes like our family doctor but she is totally useless so I knew for a fact that she'd bring us to a dead end.
His speech therapist also referred him to the hospital to have a full evaluation done. They were missing the better part of his file but, based on their own observations, pretty much ruled out ASD and diagnosed only a speech delay because he could make some eye contact. That was exactly how they said it, as if no ASD child could make any eye contact at all.
For me, I knew the EIC would be the best option as my son is not an extreme case but you need to know what to look for. These are people who work with these kids day in and day out. I fully trust their expertise. Plus, the minute we took him there, there was absolutely no denying how much more similar he was to those kids than the ones at the mainstream preschool he had been attending.
As for you, it seems a lot of the work has already been started. I expect the next step will be something like the CARS or ADOS test by a specialist to try and stimulat those behaviors and see how your daughter plays. The history will be a big chunk of it, I am sure. It won't necessarily be so long, depending on who needs to perform the test and how long the waiting list is there. At least, I am hoping it won't be. Wishing you a smooth ride that provides you with more answers than questions.