Migrate ExtensionInstalledWaiterTest into InProcessBrowserTest

Use of BrowserWithTestWindowTest is being deprecated and existing tests
requiring a Browser environment are being migrated to being browser
tests.

This CL updates the ExtensionInstalledWaiterTest fixture to an
InProcessBrowserTest and removes the BrowserWithTestWindowTest
dependency.

The test now uses the more specific extensions::ExtensionBrowserTest as
its base class to leverage its automatic setup of the extension system.
The logic for handling asynchronous operations has been updated to use
base::test::RunUntil, and browser creation/destruction now uses helpers
from InProcessBrowserTest.

Bypass-Check-License: this is a test file rename
Bug: 417766643
Change-Id: Ib76272ae88def291cffcba4d9bdbc63cf27c25fb
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6771886
Reviewed-by: Tom Lukaszewicz <tluk@chromium.org>
Auto-Submit: Mikita Kuchyn <kuchyn@chromium.org>
Commit-Queue: Mikita Kuchyn <kuchyn@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1522125}