[memories] Refactors UI handler and populates additional visit annotation

Refactors and renames the UI handler and the Service code for improved
readability. Also populates an additional visit annotation indicating
whether a visit was to the default search provider's result page. This
annotation will be used for logging histograms in a follow-up CL.

Bug: 1173908
Change-Id: I763d90d569b224bd04b73109f51ff76dc0e3d3e9
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3173778


Commit-Queue: Moe Ahmadi <mahmadi@chromium.org>
Reviewed-by: Tommy Li <tommycli@chromium.org>
Cr-Commit-Position: refs/heads/main@{#923646}