That is the typical way to start drugs. There is no good science. In other words, if you say to a psychiatrist “she only has inattention in the afternoons, plus gets shy in parties and loud music bother her” the doctor will not be able to say “oh she should try xyz because it is particularly good for those symptoms.”
Nope, there is Ritalin class and the other class. And they have different pill forms and other differences around the extras but you just need to try them to see what works for your daughter. One person with her symptoms may do well with Ritalin, and someone else with the same symptoms may prefer quialivant. No way to predict. So with that said, with extremely run of the mill, non crisis adhd as you have described, just go to a ped and start trying them out. It won’t be any different than a psych.
If your child had more complicated symptoms, like rage or anxiety or Tourette’s or bipolar, then you go to a psychiatrist because treatment will likely end up with multiple drugs and side effects will be far more important to watch.