Request types appear in the same order as you ordered them when you created your request types. That is why you need to go to Project Settings-> Request Types and reorder the request types as you would like to see them in the picker.
You can read more about request types and how to manange them here:
...word of mouth, so by 2016, we were working with several other entities on campus to implement Jira Service Desk . The Atlassian motto of “for every team” has really come true for us in this case. We...
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