Bug 1665076 - Call setSearchMode directly from search(), fix search mode handoff, and introduce new search mode probes. r=adw

This patch calls setSearchMode directly from search(). This sets up a solution for the problem in the bug and also fixes the issue where a call to search() with a restriction token would flicker the token before it was replaced with the search mode indicator. I added new tabmenu and bookmarkmenu entry points to take advantage of this new functionality.

This also fixes the issues with handoff. Besides the problem of search() recording typed for handoff, `handoff` wasn't even registered as a Telemetry probe! That was my mistake. I added a test for handoff telemetry. It was only practical to test it in PBM since it uses a different implementation than about:home that's easier to test. I wrote a lengthy comment above the subtest about why I think this is okay.

Differential Revision: https://phabricator.services.mozilla.com/D91076