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Previously web_engine_integration_tests were using default WebEngine
instance. This approach creates several issues. Particularly WebEngine
that is being tested may not be the latest verstion installed on the
test system. Updated the tests to use Launcher API to start a new
instance of WebEngine. This approach will also allow to pass custom
flags to WebEngine, which will be useful for some tests.

Also fixed initialization order in ContextProviderMain

Change-Id: Id6ffbd9a498fa42531193c5e9c6d01a932591ca7
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1938454
Commit-Queue: Wez <wez@chromium.org>
Reviewed-by: Wez <wez@chromium.org>
Auto-Submit: Sergey Ulanov <sergeyu@chromium.org>
Cr-Commit-Position: refs/heads/master@{#719851}