[OmniboxLabs] Cleans up provider and delegate

Improves code encapsulation between the provider and the delegate.
Adds checks for invariants.
Updates comments and namings.

Bug: 378538411
Change-Id: Ief181ff450baf1f3fb24503ea0602cead3723f18
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6181587
Reviewed-by: Nihar Majmudar <niharm@google.com>
Reviewed-by: Salma Elmahallawy <esalma@google.com>
Reviewed-by: Fabio Tirelo <ftirelo@chromium.org>
Commit-Queue: Moe Ahmadi <mahmadi@chromium.org>
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Cr-Commit-Position: refs/heads/main@{#1408135}