We will be interested in reading others answers to this question! We have terms for some of these things, but they are terms we made for ourselves, and may not be the "correct" or official terms. If you need to correct terms to fill out paperwork for a service dog, we agree you should ask your T to help with the forms. That is so cool you might be able to get a dog! That would be great. We have pets, and they are a great comfort to us. They make us laugh, they get us out of bed when we'd rather stay under the covers. We hope you can get the dog.
We use the term "rolling" rather than "switching" if we're changing who's at surface while co-conscious. We say "switch" if someone comes out causing lost time for others. We also have a non-human alter and if she is out, we say "change" because it's like we really become something else entirely.
Good luck! Let us know what happens!