If you want to persist the query parameters for some reports you have to do next steps:
1. Yes, if you move mouse over the report title in the main area, it changes to the editable field.
2. You can change the name.
3. You have to do save this name via left bottom "save".
4. Then it will be available in the Report drop down in the left pane.
Your saved report will be available for you only, but it is possible to copy saved report view to other user. Press see Share icon when you move mouse over the report title. See also Share Report
Note, also you can bookmark or copy and save the url from the browser instead of persisting query parameters in the report.
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