Add a File System checkbox and include the selected storage type in clearing requests. Test both checked and unchecked states.

Bug: 40227285
Change-Id: I0916ba66dca3ca9a644eba7eb51e6c01aa7ac69b
Reviewed-on: https://chromium-review.googlesource.com/c/devtools/devtools-frontend/+/8473167
Reviewed-by: Kim-Anh Tran <kimanh@chromium.org>
Reviewed-by: Simon Zünd <szuend@chromium.org>
Commit-Queue: Kim-Anh Tran <kimanh@chromium.org>